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Wrote out chapters 2-4

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Linear-Algebra-Theorems-and-Definitions.pdf
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Linear-Algebra-Theorems-and-Definitions.tex
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\newcommand{\linear}[1]{\mathcal{L}\left(#1\right)}
\newcommand{\Id}{\mathds{I}}
\newcommand{\per}[1]{\text{per}\left(#1\right)}
\newcommand{\n}[1]{\text{N}\left(#1\right)}
\newcommand{\range}[1]{\text{R}\left(#1\right)}
\newcommand{\LL}{\mathcal{L}}
\setcounter{chapter}{-1}
\author{Alexander J. Clarke}
chapter-1/linear-combinations-and-systems-of-linear-equations.tex
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\begin{definition}
\hfill\\
Let $S$ be a nonempty subset of a vector space $V$. The \textbf{span} of $S$, denoted $\text{span}(S)$, is the set consisting of all linear combinations of the vectors in $S$. For convenience, we define $\text{span}(\emptyset) = \{0\}$.
Let $S$ be a nonempty subset of a vector space $V$. The \textbf{span} of $S$, denoted $\lspan{S}$, is the set consisting of all linear combinations of the vectors in $S$. For convenience, we define $\lspan{\emptyset} = \{0\}$.
\end{definition}
\begin{theorem}
@@ -18,4 +18,4 @@
\begin{definition}
\hfill\\
A subset $S$ of a vector space $V$ \textbf{generates} (or \textbf{spans}) $V$ if $\text{span}(S) = V$. In this case, we also say that the vectors of $S$ generate (or span) $V$.
\end{definition}
\end{definition}
chapter-1/linear-dependence-and-linear-independence.tex
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\begin{definition}
\hfill\\
A subset $S$ of a vector space $V$ is called \textbf{linearly dependent} if there exist a finite number of distinct vectors $v_1, v_2, \dots, v_n$ in $S$ and scalars $a_1, a_2, \dots, a_n$ not all zero, such that
\[a_1v_2 + a_2v_2 + \dots + a_nv_n = 0\]
In this case, we also say that the vectors of $S$ are linearly dependent.\\
For any vectors $v_1, v_2, \dots, v_n$, we have $a_1v_1 + a_2v_2 + \dots + a_nv_n = 0$ if $a_1 = a_2 = \dots = a_n = 0$. We call this the \textbf{trivial representation} of $0$ as a linear combination of $v_1, v_2, \dots, v_n$. Thus, for a set tot be linearly dependent, there must exist a nontrivial representation of $0$ as a linear combination of vectors in the set. Consequently, any subset of a vector space that contains the zero vector is linearly dependent, because $0 = 1 \cdot 0$ is a nontrivial representation of $0$ as a linear combination of vectors in the set.
@@ -14,15 +14,15 @@
\begin{definition}
\hfill\\
A subset $S$ of a vector space that is not linearly dependent is called \textbf{linearly independent}. As before, we also say that the vectors of $S$ are linearly independent.\\
The following facts about linearly independent sets are true in any vector space.
\begin{enumerate}
\item The empty set is linearly independent, for linearly dependent sets must be nonempty.
\item A set consisting of a single nonzero vector is linearly independent. For if $\{v\}$ is linearly dependent, then $av = 0$ for some nonzero scalar $a$. thus
\[v = a^{-1}(av) = a^{-1}0 = 0.\]
\[v = a^{-1}(av) = a^{-1}0 = 0.\]
\item A set is linearly independent if and only if the only representations of $0$ as linear combinations of its vectors are trivial representations.
\end{enumerate}
\end{definition}
@@ -39,5 +39,5 @@
\begin{theorem}
\hfill\\
Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is linearly dependent if and only if $v \in \text{span}(S)$.
\end{theorem}
Let $S$ be a linearly independent subset of a vector space $V$, and let $v$ be a vector in $V$ that is not in $S$. Then $S \cup \{v\}$ is linearly dependent if and only if $v \in \lspan{S}$.
\end{theorem}
chapter-1/vector-spaces.tex
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for $1 \leq i \leq m$ and $1 \leq j \leq n$.
\end{definition}
\begin{definition}
\begin{definition}\label{Definition 1.7}
\hfill\\
Let $S$ be any nonempty set and $\F$ be any field, and let $\mathcal{F}(S, \F)$ denote the set of all functions from $S$ to $\F$. Two functions $f$ and $g$ in $\mathcal{F}(S, \F)$ are called \textbf{equal} if $f(s) = g(s)$ for each $s \in S$. The set $\mathcal{F}(S, \F)$ is a vector space with the operations of addition and scalar multiplication defined for $f,g \in \mathcal{F}(S, \F)$ and $c \in \F$ defined by
chapter-2/composition-of-linear-transformations-and-matrix-multiplication.tex
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\section{Compositions of Linear Transformations and Matrix Multiplication}
\begin{theorem}
\hfill\\
Let $V$, $W$, and $Z$ be vector spaces over the same field $\F$, and let $T: V \to U$ and $U: W \to Z$ be linear. Then $UT: V \to Z$ is linear.
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ be a vector space. Let $T, U_1, U_2 \in \LL(V)$. Then
\begin{enumerate}
\item $T(U_1 + U_2) = TU_1 + TU_2$ and $(U_1 + U_2)T = U_1T + U_2T$
\item $T(U_1U_2) = (TU_1)U_2$
\item $TI = IT = T$
\item $a(U_1U_2) = (aU_1)U_2 = U_1(aU_2)$ for all scalars $a$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix. We define the \textbf{product} of $A$ and $B$, denoted $AB$, to be the $m \times p$ matrix such that
\[(AB)_{ij} = \sum_{k=1}^{n}A_{ik}B_{kj}\ \ \text{for}\ \ 1 \leq i \leq m,\ \ 1 \leq j \leq p.\]
Notice that $(AB)_{ij}$ is the sum of products of corresponding entries from the $i$th row of $A$ and the $j$th column of $B$.\\
The reader should observe that in order for the product $AB$ to be defined, there are restrictions regarding the relative sizes of $A$ and $B$. The following mnemonic device is helpful: ``$(m \times n) \cdot (n \times p) = (m \times p)$"; that is, in order for the product $AB$ to be defined, the two ``inner" dimensions must be equal, and the two ``outer" dimensions yield the size of the product.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$, $W$, and $Z$ be finite-dimensional vector spaces with ordered bases $\alpha$, $\beta$, and $\gamma$, respectively. Let $T: V \to W$ and $U: W \to Z$ be linear transformations. Then
\[[UT]_\alpha^\gamma = [U]_\beta^\gamma[T]_\alpha^\beta\]
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a finite-dimensional vector space with an ordered basis $\beta$. Let $T, U \in \LL(V)$. Then $[UT]_\beta = [U]_\beta [T]_\beta$.
\end{corollary}
\begin{definition}
\hfill\\
We define the \textbf{Kronecker delta} $\delta_{ij}$ by $\delta_{ij}=1$ if $i = j$ and $\delta_{ij}=0$ if $i \neq j$. The $n \times n$ \textbf{identity matrix} $I_n$ is defined by $(I_n)_{ij} = \delta_{ij}$.
\end{definition}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix, $B$ and $C$ be $n \times p$ matrices, and $D$ and $E$ be $q \times m$ matrices. Then
\begin{enumerate}
\item $A(B + C) = AB + AC$ and $(D + E)A = DA + EA$.
\item $a(AB) = (aA)B = A(aB)$ for any scalar $a$.
\item $I_mA = A = AI_n$.
\item If $V$ is an $n$-dimensional vector space with an ordered basis $\beta$, then $[I_V]_\beta = I_n$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $A$ be an $m \times n$ matrix, $B_1, B_2, \dots, B_k$ be $n \times p$ matrices, $C_1, C_2, \dots, C_k$ be $q \times m$ matrices, and $a_1, a_2, \dots, a_k$ be scalars. Then
\[A\left(\sum_{i=1}^{k}a_iB_i\right) = \sum_{i=1}^{k}a_iAB_i\]
and
\[\left(\sum_{i=1}^{k}a_iC_i\right)A = \sum_{i=1}^{k}a_iC_iA.\]
\end{corollary}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix and $B$ be an $n \times p$ matrix. For each $j$ ($1 \leq j \leq p$) let $u_j$ and $v_j$ denote the $j$th columns of $AB$ and $B$, respectively. Then
\begin{enumerate}
\item $u_j = Av_j$.
\item $v_j = Be_j$, where $e_j$ is the $j$th standard vector of $\F^p$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces having ordered bases $\beta$ and $\gamma$, respectively, and let $T: V \to W$ be linear. Then, for each $u \in V$, we have
\[[T(u)]_\gamma = [T]_\beta^\gamma [u]_\beta.\]
\end{theorem}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix with entries from a field $\F$. We denote $L_A: \F^n \to \F^m$ defined by $L_A(x) = Ax$ (the matrix product of $A$ and $x$) for each column vector $x \in \F^n$. We call $L_A$ a \textbf{left-multiplication transformation}.
\end{definition}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix with entries from $\F$. Then the left-multiplication transformation $L_A: \F^n \to \F^m$ is linear. Furthermore, if $B$ is any other $m \times n$ matrix (with entries from $\F$) and $\beta$ and $\gamma$ are the standard ordered bases for $\F^n$ and $\F^m$, respectively, then we have the following properties.
\begin{enumerate}
\item $[L_A]_\beta^\gamma = A$.
\item $L_A = L_B$ if and only if $A = B$.
\item $L_{A + B} = L_A + L_B$ and $L_{aA} = aL_A$ for all $a \in \F$.
\item If $T: \F^n \to \F^m$ is linear, then there exists a unique $m \times n$ matrix $C$ such that $T = L_C$. In fact, $C = [T]_\beta^\gamma$.
\item If $E$ is an $n \times p$ matrix, then $L_{AE} = L_AL_E$.
\item If $m = n$, then $L_{I_n} = I_{\F^n}$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $A$, $B$, and $C$ be matrices such that $A(BC)$ is defined. Then $(AB)C$ is also defined and $A(BC)=(AB)C$; that is, matrix multiplication is associative.
\end{theorem}
\begin{definition}
\hfill\\
An \textbf{incidence matrix} is a square matrix in which all the entries are either zero or one and, for convenience, all the diagonal entries are zero. If we have a relationship on a set of $n$ objects that we denote $1, 2, \dots, n$, then we define the associated incidence matrix $A$ by $A_{ij} = 1$ if $i$ is related to $j$, and $A_{ij} = 0$ otherwise.
\end{definition}
\begin{definition}
\hfill\\
A relationship among a group of people is called a \textbf{dominance relation} if the associated incidence matrix $A$ has the property that for all distinct pairs $i$ and $j$, $A_{ij} = 1$ if and only if $A_{ji} = 0$, that is, given any two people, exactly one of them \textit{dominates} the other.
\end{definition}
chapter-2/dual-spaces.tex
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\section{Dual Spaces}
\begin{definition}
\hfill\\
A linear transformation from a vector space $V$ into its field of scalars $\F$, which is itself a vector space of dimension 1 over $\F$, is called a \textbf{linear functional} on $V$. We generally use the letters $\mathsf{f}, \mathsf{g}, \mathsf{h}, \dots$. to denote linear functionals.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ be a vector space of continuous real-valued functions on the interval $[0, 2\pi]$. Fix a function $g \in V$. The function $\mathsf{h}: V \to \R$, defined by
\[\mathsf{h}(x) = \frac{1}{2\pi} \int_{0}^{2\pi}x(t)g(t) dt\]
is a linear functional on $V$. In the cases that $g(t)$ equals $\sin(nt)$ or $\cos (nt)$, $\mathsf{h}(x)$ is often called the \textbf{\textit{n}th Fourier coefficient of $x$}.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ be a finite dimensional vector space, and let $\beta = \{x_1, x_2, \dots, x_n\}$ be an ordered basis for $V$. For each $i = 1, 2, \dots, n$, define $\mathsf{f}_i(x) = a_i$, where
\[[x]_\beta = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}\]
is the coordinate vector of $x$ relative to $\beta$. Then $\mathsf{f}$ is a linear function on $V$ called the \textbf{\textit{i}th coordinate function with respect to the basis $\beta$}. Note that $\mathsf{f}_i(x_j) = \delta_{ij}$, where $\delta_{ij}$ is the Kronecker delta. These linear functionals play an important role in the theory of dual spaces (see \autoref{Theorem 2.24}).
\end{definition}
\begin{definition}
\hfill\\
For a vector space $V$ over $\F$, we define the \textbf{dual space} of $V$ to be the vector space $\LL(V, \F)$, denoted by $V^*$.\\
Thus $V^*$ is the vector space consisting of all linear functionals on $V$ with the operations of addition and scalar multiplication. Note that if $V$ is finite-dimensional, then by \autoref{Corollary 2.7}
\[\ldim{V^*}= \ldim{\LL(V,\F)} = \ldim{V} \cdot \ldim{\F} = \ldim{V}.\]
Hence by \autoref{Theorem 2.19}, $V$ and $V^*$ are isomorphic. We also define the \textbf{double dual} $V^{**}$ of $V$ to be the dual of $V^*$. In \autoref{Theorem 2.26}, we show, in fact, that there is a natural identification of $V$ and $V^{**}$ in the case that $V$ is finite-dimensional.
\end{definition}
\begin{theorem}\label{Theorem 2.24}
\hfill\\
Suppose that $V$ is a finite-dimensional vector space with the ordered basis $\beta = \{x_1, x_2, \dots, x_n\}$. Let $\mathsf{f}_i$ ($1 \leq i \leq n$) be the $i$th coordinate function with respect to $\beta$ as just defined, and let $\beta^*=\{\mathsf{f}_1, \mathsf{f}_2, \dots, \mathsf{f}_n\}$. Then $\beta^*$ is an ordered basis for $V^*$, and, for any $\mathsf{f} \in V^*$, we have
\[\mathsf{f} = \sum_{i=1}^{n}\mathsf{f}(x_i)\mathsf{f}_i.\]
\end{theorem}
\begin{definition}
\hfill\\
Using the notation of \autoref{Theorem 2.24}, we call the ordered basis $\beta^* = \{\mathsf{f}_1, \mathsf{f}_2, \dots, \mathsf{f}_n\}$ of $V^*$ that satisfies $\mathsf{f}_i(x_j) = \delta_{ij}$ ($1 \leq i,\ j \leq n$) the \textbf{dual basis} of $\beta$.
\end{definition}
\begin{theorem}\label{Theorem 2.25}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces over $\F$ with ordered bases $\beta$ and $\gamma$, respectively. For any linear transformation $T: V \to W$, the mapping $T^t: W^* \to V^*$ defined by $T^t(\mathsf{g}) = \mathsf{g}T$ for all $\mathsf{g} \in W^*$ is a linear transformation with the property that $[T^t]_{\gamma^*}^{\beta^*} = ([T]_\beta^\gamma)^t$.
\end{theorem}
\begin{definition}
\hfill\\
The linear transformation $T^t$ defined in \autoref{Theorem 2.25} is called the \textbf{transpose} of $T$. It is clear that $T^t$ is the unique linear transformation $U$ such that $[U]_{\gamma^*}^{\beta^*} = ([T]_\beta^\gamma)^t$.
\end{definition}
\begin{definition}
\hfill\\
For a vector $x$ in a finite-dimensional vector space $V$, we define the linear functional $\hat{x}: V^* \to \F$ on $V^*$ by $\hat{x}(\mathsf{f}) = \mathsf{f}(x)$ for every $\mathsf{f} \in V^*$. Since $\hat{x}$ is a linear functional on $V^*$, $\hat{x} \in V^{**}$.\\
The correspondence $x \leftrightarrow \hat{x}$ allows us to define the desired isomorphism between $V^*$ and $V^{**}$.
\end{definition}
\begin{lemma}
\hfill\\
Let $V$ be a finite-dimensional vector space, and let $x \in V$. If $\hat{x}(\mathsf{f})=0$ for all $\mathsf{f} \in V^*$, then $x = 0$.
\end{lemma}
\begin{theorem}\label{Theorem 2.26}
\hfill\\
Let $V$ be a finite-dimensional vector space, and define $\psi: V \to V^{**}$ by $\psi(x) = \hat{x}$. Then $\psi$ is an isomorphism.
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a finite-dimensional vector space with dual space $V^*$. Then every ordered basis for $V^*$ is the dual basis for some basis $V$.
\end{corollary}
\begin{definition}
\hfill\\
Let $V$ be a finite-dimensional vector space over $\F$. For every subset $S$ of $V$, define the \textbf{annihilator} $S^0$ of $S$ as
\[S^0 = \{\mathsf{f} \in V^*\ |\ \mathsf{f}(x) = 0,\ \text{for all}\ x \in S\}\]
\end{definition}
chapter-2/homogeneous-linear-differential-equations-with-constant-coefficients.tex
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\section{Homogeneous Linear Differential Equations with Constant Coefficients}
\begin{definition}
\hfill\\
A \textbf{differential equation} in an unknown function $y = y(t)$ is an equation involving $y$, $t$, and derivatives of $y$. If the differential equation is of the form
\begin{equation}
a_ny^{(n)}+a_{n-1}y^{(n-)} + \dots + a_1y^{(1)}+a_0y = f,
\end{equation}
where $a_0, a_1, \dots, a_n$ and $f$ are functions of $t$ and $y^{(k)}$ denotes the $k$th derivative of $y$, then the equation is said to be \textbf{linear}. The functions $a_i$ are called the \textbf{coefficients} of the differential equation. When $f$ is identically zero, (2.1) is called \textbf{homogeneous}.\\
If $a_n \neq 0$, we say that differential equation (2.1) is of \textbf{order \textit{n}}. In this case, we divide both sides by $a_n$ to obtain a new, but equivalent, equation
\[y^{(n)} + b_{n-1}y^{(n-1)} + \dots + b_1y^{(1)} + b_0y = 0,\]
where $b_i = a_i/a_n$ for $i=0, 1, \dots, n-1$. Because of this observation, we always assume that the coefficient $a_n$ in (2.1) is $1$.\\
A \textbf{solution} to (2.1) is a function that when substituted for $y$ reduces (2.1) to an identity.
\end{definition}
\begin{definition}
\hfill\\
Given a complex-valued function $x \in \mathcal{F}(\R, \C)$ of a real variable $t$ (where $\mathcal{F}(\R, \C)$ is the vector space defined in \autoref{Definition 1.7}), there exist unique real-valued functions $x_1$ and $x_2$ of $t$, such that
\[x(t) = x_1(t) + ix_2(t)\ \ \ \text{for}\ \ \ t \in \R,\]
where $i$ is the imaginary number such that $i^2 = -1$. We call $x_1$ the \textbf{real part} and $x_2$ the \textbf{imaginary part} of $x$.
\end{definition}
\begin{definition}
\hfill\\
Given a function $x \in \mathcal{F}(\R, \C)$ with real part $x_1$ and imaginary part $x_2$, we say that $x$ is \textbf{differentiable} if $x_1$ and $x_2$ are differentiable. If $x$ is differentiable, we define the \textbf{derivative} $x'$ of $x$ by
\[x' = x'_1 + ix'_2\]
\end{definition}
\begin{theorem}
\hfill\\
Any solution to a homogeneous linear differential equation with constant coefficients has derivatives of all orders; that is, if $x$ is a solution to such an equation, then $x^(k)$ exists for every positive integer $k$.
\end{theorem}
\begin{definition}
\hfill\\
We use $\C^\infty$ to denote the set of all functions in $\mathcal{F}(\R, \C)$ that have derivatives of all orders.
\end{definition}
\begin{definition}
\hfill\\
For any polynomial $p(t)$ over $\C$ of positive degree, $p(D)$ is called a \textbf{differential operator}. The \textbf{order} of the differential operator $p(D)$ is the degree of the polynomial $p(t)$.
\end{definition}
\begin{definition}
\hfill\\
Given the differential equation
\[y^{(n)} + a_{n-1}y^{(n-1)}+ \dots + a_1y^{(1)} + a_0y = 0,\]
the complex polynomial
\[p(t) = t^n + a_{n-1}t^{n-1} + \dots + a_1t + a_0\]
is called the \textbf{auxiliary polynomial} associated with the equation.
\end{definition}
\begin{theorem}
\hfill\\
The set of all solutions to a homogeneous linear differential equation with constant coefficients coincides with the null space of $p(D)$ where $p(t)$ is the auxiliary polynomial associated with the equation.
\end{theorem}
\begin{corollary}
\hfill\\
The set of all solutions to a homogeneous linear differential equation with constant coefficients is a subspace of $\C^\infty$.
\end{corollary}
\begin{definition}
\hfill\\
We call the set of solutions to a homogeneous linear differential equation with constant coefficients the \textbf{solution space} of the equation.
\end{definition}
\begin{definition}
\hfill\\
Let $c = a+ib$ be a complex number with real part $a$ and imaginary part $b$. Define
\[e^c = e^a(\cos(b) + i\sin(b)).\]
The special case
\[e^{ib} = \cos(b) + i\sin(a)\]
is called \textbf{Euler's formula}.
\end{definition}
\begin{definition}
\hfill\\
A function $f: \R \to \C$ defined by $f(t) = e^{ct}$ for a fixed complex number $c$ is called an \textbf{exponential function}.
\end{definition}
\begin{theorem}
\hfill\\
For any exponential function $f(t) = e^{ct}$, $f'(t) = ce^{ct}$.
\end{theorem}
\begin{theorem}
\hfill\\
Recall that the \textbf{order} of a homogeneous linear differential equation is the degree of its auxiliary polynomial. Thus, an equation of order 1 is of the form
\begin{equation}
y' + a_0y = 0.
\end{equation}
The solution space for (2.2) is of dimension 1 and has $\{e^{-a_0t}\}$ as a basis.
\end{theorem}
\begin{corollary}
\hfill\\
For any complex number $c$, the null space of the differential operator $D-c\mathsf{l}$ has $\{e^{ct}\}$ as a basis.
\end{corollary}
\begin{theorem}
\hfill\\
Let $p(t)$ be the auxiliary polynomial for a homogeneous linear differential equation with constant coefficients. For any complex number $c$, if $c$ is a zero of $p(t)$, then $e^{ct}$ is a solution to the differential equation.
\end{theorem}
\begin{theorem}
\hfill\\
For any differential operator $p(D)$ of order $n$, the null space of $p(D)$ is an $n$-dimensional subspace of $\C^\infty$.
\end{theorem}
\begin{lemma}
\hfill\\
The differential operator $D - c\mathsf{l}: \C^\infty \to \C^\infty$ is onto for any complex number $c$.
\end{lemma}
\begin{lemma}
\hfill\\
Let $V$ be a vector space, and suppose that $T$ and $U$ are linear operators on $V$ such that $U$ is onto and the null spaces of $T$ and $U$ are finite-dimensional. Then the null space of $TU$ is finite-dimensional, and
\[\ldim{\n{TU}} = \ldim{\n{T}} + \ldim{\n{U}}\]
\end{lemma}
\begin{corollary}
\hfill\\
The solution space of any $n$th-order homogeneous linear differential equation with constant coefficients is an $n$-dimensional subspace of $\C^\infty$.
\end{corollary}
\begin{theorem}
\hfill\\
Given $n$ distinct complex numbers $c_1, c_2, \dots, c_n$, the set of exponential functions $\{e^{c_1t},e^{c_2t},\dots,e^{c_nt}\}$ is linearly independent.
\end{theorem}
\begin{corollary}
\hfill\\
For any $n$th-order homogeneous linear differential equation with constant coefficients, if the auxiliary polynomial has $n$ distinct zeros $c_1, c_2, \dots, c_n$, then $\{e^{c_1t}, e^{c_2t}, \dots, e^{c_nt}\}$ is a basis for the solution space of the differential equation.
\end{corollary}
\begin{lemma}
\hfill\\
For a given complex number $c$ and a positive integer $n$, suppose that $(t-c)^n$ is the auxiliary polynomial of a homogeneous linear differential equation with constant coefficients. Then the set
\[\beta = \{e^{ct}, te^{ct}, \dots, t^{n-1}e^{ct}\}\]
is a basis for the solution space of the equation.
\end{lemma}
\begin{theorem}
\hfill\\
Given a homogeneous linear differential equation with constant coefficients and auxiliary polynomial
\[(t-c_1)^{n_1}(t-c_2)^{n_2}\dots(t-c_k)^{n_k},\]
where $n_1, n_2, \dots, n_k$ are positive integers and $c_1, c_2, \dots, c_k$ are distinct complex numbers, the following set is a basis for the solution space of the equation:
\[\{e^{c_1t}, te^{c_1t},\dots, t^{n_1-1}e^{c_1t}, \dots, e^{c_kt}, te^{c_kt}, \dots, t^{n_k-1}e^{c_kt}\}\]
\end{theorem}
\begin{definition}
\hfill\\
A differential equation
\[y^{(n)} + a_{n-1}y^{(n-1)} + \dots + a_1y^{(1)} + a_0y = x\]
is called a \textbf{nonhomogeneous} linear differential equation with constant coefficients if the $a_i$'s are constant and $x$ is a function that is not identically zero.
\end{definition}
chapter-2/invertibility-and-isomorphisms.tex
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\section{Invertibility and Isomorphisms}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. A function $U: W \to V$ is said to be an \textbf{inverse} of $T$ if $TU = I_W$ and $UT = I_V$. If $T$ has an inverse, then $T$ is said to be \textbf{invertible}. If $T$ is invertible, then the inverse of $T$ is unique and is denoted by $T^{-1}$.\\
The following facts hold for invertible functions $T$ and $U$.
\begin{enumerate}
\item $(TU)^{-1} = U^{-1}T^{-1}$.
\item $(T^{-1})^{-1} = T$; in particular, $T^{-1}$ is invertible.
\end{enumerate}
We often use the fact that a function is invertible if and only if it is one-to-one and onto. We can therefore restate \autoref{Theorem 2.5} as follows:
\begin{enumerate}
\setcounter{enumi}{2}
\item Let $T: V \to W$ be a linear transformation, where $V$ and $W$ are finite-dimensional vector spaces of equal dimension. then $T$ is invertible if and only if $\rank{T} = \ldim{T}$.
\end{enumerate}
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear and invertible. Then $T^{-1}: W \to V$ is linear.
\end{theorem}
\begin{definition}
\hfill\\
Let $A$ be an $n \times n$ matrix. Then $A$ is \textbf{invertible} if there exists an $n \times n$ matrix $B$ such that $AB = BA = I$.\\
If $A$ is invertible, then the matrix $B$ such that $AB = BA = I$ is unique. (If $C$ were another such matrix, then $C = CI = C(AB) = (CA)B = IB = B$.) The matrix $B$ is called the \textbf{inverse} of $A$ and is denoted by $A^{-1}$.
\end{definition}
\begin{lemma}
\hfill\\
Let $T$ be an invertible linear transformation from $V$ to $W$. Then $V$ is finite-dimensional if and only if $W$ is finite-dimensional. In this case, $\ldim{V} = \ldim{W}$
\end{lemma}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces with ordered bases $\beta$ and $\gamma$, respectively. Let $T: V \to W$ be linear. Then $T$ is invertible if and only if $[T]_\beta^\gamma$ is invertible. Furthermore, $[T^{-1}]_\gamma^\beta = ([T]_\beta^\gamma)^{-1}$.
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a finite-dimensional vector space with an ordered bases $\beta$, and let $T: V \to V$ be linear. Then $T$ is invertible if and only if $[T]_\beta$ is invertible. Furthermore, $[T^{-1}]_\beta = ([T]_\beta)^{-1}$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $A$ be and $n \times n$ matrix. Then $A$ is invertible if and only if $L_A$ is invertible. Furthermore, $(L_A)^{-1} = L_{A^{-1}}$.
\end{corollary}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces. We say that $V$ is \textbf{isomorphic} to $W$ if there exists a linear transformation $T: V \to W$ that is invertible. Such a linear transformation is called an \textbf{isomorphism} from $V$ onto $W$.
\end{definition}
\begin{theorem}\label{Theorem 2.19}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces (over the same field). Then $V$ is isomorphic to $W$ if and only if $\ldim{V} = \ldim{W}$.
\end{theorem}
\begin{corollary}
\hfill\\
Let $V$ be a vector space over $\F$. Then $V$ is isomorphic to $\F^n$ if and only if $\ldim{V} = n$.
\end{corollary}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces over $\F$ of dimensions $n$ and $m$, respectively, and let $\beta$ and $\gamma$ be ordered bases for $V$ and $W$, respectively. Then the function $\Phi: \LL(V,W) \to M_{m \times n}(\F)$, defined by $\Phi(T) = [T]_\beta^\gamma$ for $T \in \LL(V,W)$ is an isomorphism.
\end{theorem}
\begin{corollary}\label{Corollary 2.7}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces of dimension $n$ and $m$, respectively. Then $\LL(V,W)$ is finite-dimensional of dimension $mn$.
\end{corollary}
\begin{definition}
\hfill\\
Let $\beta$ be an ordered basis for an $n$-dimensional vector space $V$ over the field $\F$. The \textbf{standard representation of $V$ with respect to $\beta$} is the function $\phi_\beta: V \to \F^n$ defined by $\phi_\beta(x) = [x]_\beta$ for each $x \in V$.
\end{definition}
\begin{theorem}
\hfill\\
For any finite-dimensional vector space $V$ with ordered basis $\beta$, $\phi_\beta$ is an isomorphism.
\end{theorem}
chapter-2/linear-transformations-null-spaces-and-ranges.tex
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\section{Linear Transformations, Null Spaces, and Ranges}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces (over $\F$). We call a function $T: V \to W$ a \textbf{linear transformation from $V$ to $W$} if, for all $x,y \in V$, and $c \in \F$, we have
\begin{enumerate}
\item $T(x + y) = T(x) + T(y)$, and
\item $T(cx) = cT(x)$
\end{enumerate}
If the underlying field $\F$ is the field of rational numbers, then (1) implies (2), but, in general (1) and (2) are logically independent.\\
We often simply call $T$ \textbf{linear}.
\end{definition}
\begin{remark}
\hfill\\
Let $V$ and $W$ be vector spaces (over $\F$). Let $T: V \to W$ be a linear transformation. Then the following properties hold:
\begin{enumerate}
\item If $T$ is linear, then $T(0) = 0$.
\item $T$ is linear if and only if $T(cx + y) = cT(x) + T(y)$ for all $x,y \in V$ and $c \in \F$.
\item If $T$ is linear, then $T(x-y)=T(x)-T(y)$ for all $x,y \in V$.
\item $T$ is linear if and only if, for $x_1, x_2, \dots, x_n \in V$ and $a_1, a_2, \dots, a_n \in \F$, we have
\[T\left(\sum_{i=1}^{n}a_ix_i\right)=\sum_{i=1}^{n}a_iT(x_i).\]
\end{enumerate}
We generally use property 2 to prove that a given transformation is linear.
\end{remark}
\begin{definition}
\hfill\\
For any angle $\theta$, define $T_\theta: \R^2 \to \R^2$ by the rule: $T_\theta(a_1, a_2)$ is the vector obtained by rotating $(a_1, a_2)$ counterclockwise by $\theta$ if $(a_1, a_2) \neq (0, 0)$, and $T_\theta(0,0) = (0,0)$. Then $T_\theta: \R^2 \to \R^2$ is a linear transformation that is called the \textbf{rotation by $\theta$}.
\end{definition}
\begin{definition}
\hfill\\
Define $T: \R^2 \to \R^2$ by $T(a_1, a_2) = (a_1, -a_2)$. $T$ is called the \textbf{reflection about the \textit{x}-axis}.
\end{definition}
\begin{definition}
\hfill\\
For vector spaces $V$ and $W$ (over $\F$), we define the \textbf{identity transformation} $I_V: V \to V$ by $I_V(x) = x$ for all $x \in V$.\\
We define the \textbf{zero transformation} $T_0: V \to W$ by $T_0(x) = 0$ for all $x \in V$.\\
\textbf{Note:} We often write $I$ instead of $I_V$.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. We define the \textbf{null space} (or \textbf{kernel}) $\n{T}$ to be the set of all vectors $x \in V$ such that $T(x)=0$; that is, \\$\n{T} = \{x \in V\ |\ T(x) = 0\}$.
We define the \textbf{range} (or \textbf{image}) $\range{T}$ of $T$ to be the subset of $W$ consisting of all images (under $T$) of vectors in $V$; that is, $\range{T} = \{T(x)\ |\ x \in V\}$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces and $T: V \to W$ be linear. Then $\n{T}$ and $\range{T}$ are subspaces of $V$ and $W$, respectively.
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. If $\beta = \{v_1, v_2, \dots, v_n\}$ is a basis for $V$, then
\[\range{T} = \lspan{T(\beta)} = \lspan{\{T(v_1), T(v_2), \dots, T(v_n)\}}.\]
\end{theorem}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. If $\n{T}$ and $\range{T}$ are finite-dimensional, then we define the \textbf{nullity} of $T$, denoted $\nullity{T}$, and the \textbf{rank} of $T$, denoted $\rank{T}$, to be the dimensions of $\n{T}$ and $\range{T}$, respectively.
\end{definition}
\begin{theorem}[\textbf{Dimension Theorem}]
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. If $V$ is finite-dimensional, then
\[\nullity{T} + \rank{T} = \ldim{V}\]
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces, and let $T: V \to W$ be linear. Then $T$ is one-to-one if and only if $\n{T} = \{0\}$.
\end{theorem}
\begin{theorem}\label{Theorem 2.5}
\hfill\\
Let $V$ and $W$ be vector spaces of equal (finite) dimension, and let $T: V \to W$ be linear. Then the following are equivalent.
\begin{enumerate}
\item $T$ is one-to-one.
\item $T$ is onto.
\item $\rank{T} = \ldim{V}$.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces over $\F$, and suppose that $\{v_1, v_2, \dots, v_n\}$ is a basis for $V$. For $w_1, w_2, \dots, w_n$ in $W$, there exists exactly one linear transformation $T: V \to W$ such that $T(v_i) = w_i$ for $i = 1, 2, \dots, n$.
\end{theorem}
\begin{corollary}\label{Corollary 2.1}
\hfill\\
Let $V$ and $W$ be vector spaces, and suppose that $V$ has a finite basis $\{v_1, v_2, \dots, v_n\}$. If $U,T: V \to W$ are linear and $U(v_i) = T(v_i)$, for $i = 1, 2, \dots, n$, then $U = T$.
\end{corollary}
\begin{definition}
\hfill\\
Let $V$ be a vector space and $W_1$ and $W_2$ be subspaces of $V$ such that $V = W_1 \oplus W_2$. A function $T: V \to V$ is called the \textbf{projection on $W_1$ along $W_2$} if, for $x = x_1 + x_2$ with $x_1 \in W$ and $x_2 \in W_2$, we have $T(x) = x_1$.
\end{definition}
\begin{definition}
\hfill\\
Let $V$ be a vector space, and let $T: V \to W$ be linear. A subspace $W$ of $V$ is said to be \textbf{$T$-invariant} if $T(x) \in W$ for every $x \in W$, that is, $T(W) \subseteq W$. If $W$ is $T$-invariant, we define the \textbf{restriction of $T$ on $W$} to be the function $T_W: W \to W$ defined by $T_W(x) = T(x)$ for all $x \in W$.
\end{definition}
chapter-2/the-change-of-coordinate-matrix.tex
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\section{The Change of Coordinate Matrix}
\begin{theorem}\label{Theorem 2.22}
\hfill\\
Let $\beta$ and $\beta'$ be two ordered bases for a finite-dimensional vector pace $V$, and let $Q = [I_V]_{\beta'}^\beta$. Then
\begin{enumerate}
\item $Q$ is invertible.
\item For any $v \in V$, $[v]_\beta = Q[v]_{\beta'}$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
The matrix $Q=[I_V]_{\beta'}^\beta$, defined in \autoref{Theorem 2.22}, is called a \textbf{change of coordinate matrix}. Because of part (2) of the theorem, we say that \textbf{$Q$ changes $\beta'$-coordinates into $\beta$-coordinates}.
\end{definition}
\begin{definition}
\hfill\\
A linear transformation that maps a vector space $V$ into itself is called a \textbf{linear operator on $V$}.
\end{definition}
\begin{theorem}
\hfill\\
Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $\beta$ and $\beta'$ be ordered bases for $V$. Suppose that $Q$ is the change of coordinate matrix that changes $\beta'$-coordinates into $\beta$-coordinates. Then
\[[T]_{\beta'}=Q^{-1}[T]_\beta Q\]
\end{theorem}
\begin{corollary}
\hfill\\
Let $A \in M_{n \times n}(\F)$, and let $\gamma$ be an ordered basis for $\F^n$. Then $[L_A]_\gamma = Q^{-1}AQ$, where $Q$ is the $n \times n$ matrix whose $j$th column is the $j$th vector of $\gamma$.
\end{corollary}
\begin{definition}
\hfill\\
Let $A$ and $B$ be matrices in $M_{n \times n}(\F)$. We say that $B$ is \textbf{similar} to $A$ if there exists an invertible matrix $Q$ such that $B = Q^{-1}AQ$.\\
Notice that the relation of similarity is an equivalence relation. So we need only say that $A$ and $B$ are similar.
\end{definition}
chapter-2/the-matrix-representation-of-a-linear-transformation.tex
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\section{The Matrix Representation of a Linear Transformation}
\begin{definition}
\hfill\\
Let $V$ be a finite-dimensional vector space. An \textbf{ordered basis} for $V$ is a basis for $V$ endowed with a specific order; that is, an ordered basis for $V$ is a finite sequence of linearly independent vectors in $V$ that generates $V$.\\
For the vector space $\F^n$, we call $\{e_1, e_2, \dots, e_n\}$ the \textbf{standard ordered basis} for $\F^n$. Similarly, for the vector space $P_n(\F)$, we call $\{1, x, \dots, x^n\}$ the \textbf{standard ordered basis} for $P_n(\F)$.
\end{definition}
\begin{definition}
\hfill\\
Let $\beta = \{v_1, v_2, \dots, v_n\}$ be an ordered basis for a finite-dimensional vector space $V$. For $x \in V$, let $a_1, a_2, \dots, a_n$ be the unique scalar values such that
\[x = \sum_{i=1}^{n}a_iv_i.\]
We define the \textbf{coordinate vector of $x$ relative to $\beta$}, denoted by $[x]_\beta$, by
\[[x]_\beta = \begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n\end{pmatrix}.\]
Notice that $[v_i]_\beta = e_i$ in the preceding definition. It can be shown that the correspondence $x \to [x]_\beta$ provides us with a linear transformation from $V$ to $\F^n$.
\end{definition}
\begin{notation}
\hfill\\
The following notation is used to construct a matrix representation of a linear transformation in the following definition.\\
Suppose that $V$ and $W$ are finite-dimensional vector spaces with ordered bases $\beta = \{v_1, v_2, \dots, v_n\}$ and $\gamma = \{w_1, w_2, \dots, w_m\}$, respectively. Let $T: V \to W$ be linear. Then for each $j$, $1 \leq j \leq n$, there exist unique scalars $a_{ij} \in \F$, $1 \leq i \leq m$, such that
\[T(v_j) = \sum_{i=1}^{m}a_{ij}w_i\ \ \text{for}\ 1 \leq j \leq n.\]
\end{notation}
\begin{definition}
\hfill\\
Using the notation above, we call the $m \times n$ matrix $A$ defined by $A_{ij} = a_{ij}$ the \textbf{matrix representation of $T$ in the ordered bases $\beta$ and $\gamma$.} and write $A = [T]_\beta^\gamma$. If $V = W$ and $\beta = \gamma$, then we write $A = [T]_\beta$.
Notice that the $j$th column of $A$ is simply $[T(v_j)]_\gamma$. Also observe that if $U: V \to W$ is a linear transformation such that $[U]_\beta^\gamma = [T]_\beta^\gamma$, then $U=T$ by the corollary to Theorem 2.6 (\autoref{Corollary 2.1}).
\end{definition}
\begin{definition}
\hfill\\
Let $T,U: V \to W$ be arbitrary functions, where $V$ and $W$ are vector spaces over $\F$, and let $a \in \F$. We define $T + U: V \to W$ by $(T+U)(x) = T(x) + U(x)$ for all $x \in V$, and $aT: V \to W$ by $(aT)(x) = aT(x)$ for all $x \in V$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be vector spaces over a field $\F$, and let $T,U: V \to W$ be linear.
\begin{enumerate}
\item For all $a \in \F$, $aT+U$ is linear.
\item Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations from $V$ to $W$ is a vector space over $\F$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $V$ and $W$ be vector spaces over $\F$. We denote the vector space of all linear transformations from $V$ to $W$ by $\LL(V, W)$. In the case that $V = W$, we write $\LL(V)$ instead of $\LL(V, W)$.
\end{definition}
\begin{theorem}
\hfill\\
Let $V$ and $W$ be finite-dimensional vector spaces with ordered bases $\beta$ and $\gamma$, respectively, and let $T,U: V \to W$ be linear transformations. Then
\begin{enumerate}
\item $[T+U]_\beta^\gamma = [T]_\beta^\gamma + [U]_\beta^\gamma$ and
\item $[aT]_\beta^\gamma = a[T]_\beta^\gamma$ for all scalars $a$.
\end{enumerate}
\end{theorem}
chapter-3/elementary-matrix-operations-and-elementary-matrices.tex
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\section{Elementary Matrix Operations and Elementary Matrices}
\begin{definition}
\hfill\\
Let $A$ be an $m \times n$ matrix. Any one of the following three operations on the rows [columns] of $A$ is called an \textbf{elementary row [column] operation}:
\begin{enumerate}
\item interchanging any two rows [columns] of $A$;
\item multiplying any row [column] of $A$ by a nonzero scalar;
\item adding any scalar multiple of a row [column] of $A$ to another row [column].
\end{enumerate}
Any of these three operations are called an \textbf{elementary operation}. Elementary operations are of \textbf{type 1}, \textbf{type 2}, or \textbf{type 3} depending on whether they are obtained by (1), (2), or (3).
\end{definition}
\begin{definition}
\hfill\\
An $n \times n$ \textbf{elementary matrix} is a matrix obtained by performing an elementary operation on $I_n$. The elementary matrix is said to be of \textbf{type 1}, \textbf{2}, or \textbf{3} according to whether the elementary operation performed on $I_n$ is a type 1, 2, or 3 operation, respectively.
\end{definition}
\begin{theorem}
\hfill\\
Let $A \in M_{m \times n}(\F)$, and suppose that $B$ is obtained from $A$ by performing an elementary row [column] operation. Then there exists an $m \times m$ [$n \times n$] elementary matrix $E$ such that $B = EA$ [$B = AE]$. In fact, $E$ is obtained from $I_m$ [$I_n]$ by performing the same elementary row [column] operation as that which was performed on $A$ to obtain $B$. Conversely, if $E$ is an elementary $m \times m$ [$n \times n$] matrix, then $EA$ [$AE$] is the matrix obtained from $A$ by performing the same elementary row [column] operation as that which produces $E$ from $I_m$ [$I_n$].
\end{theorem}
\begin{theorem}
\hfill\\
Elementary matrices are invertible, and the inverse of an elementary matrix is an elementary matrix of the same type.
\end{theorem}
chapter-3/systems-of-linear-equations-computational-aspects.tex
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\section{Systems of Linear Equations -- Computational Aspects}
\begin{definition}
\hfill\\
Two systems of linear equations are called \textbf{equivalent} if they have the same solution set.
\end{definition}
\begin{theorem}
\hfill\\
Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns, and let $C$ be an invertible $m \times n$ matrix. Then the system $(CA)x = Cb$ is equivalent to $Ax = b$.
\end{theorem}a
\begin{corollary}
\hfill\\
Let $Ax = b$ be a system of $m$ linear equations in $n$ unknowns. If $(A'|b')$ is obtained from $(A|b)$ by a finite number of elementary row operations, then the system $A'x = b'$ is equivalent to the original system.
\end{corollary}
\begin{definition}
\hfill\\
A matrix is said to be in \textbf{reduced row echelon form} if the following three conditions are satisfied.
\begin{enumerate}
\item Any row containing a nonzero entry precedes any row in which all the entries are zero (if any).
\item The first nonzero entry in each row is the only nonzero entry in its column.
\item The first nonzero entry in each row is 1 and it occurs in a column to the right of the first nonzero entry in the preceding row.
\end{enumerate}
\end{definition}
\begin{definition}
\hfill\\
The following procedure for reducing an augmented matrix to reduced row echelon form is called \textbf{Gaussian elimination}. It consists of two separate parts.
\begin{enumerate}
\item In the \textit{forward pass}, the augmented matrix is transformed into an upper triangular matrix in which the first nonzero entry of each row is $1$, and it occurs in a column to the right of the first nonzero entry in the preceding row.
\item In the \textit{backward pass} or \textit{back-substitution}, the upper triangular matrix is transformed into reduced row echelon form by making the first nonzero entry of each row the only nonzero entry of its column.
\end{enumerate}
\end{definition}
\begin{theorem}
\hfill\\
Gaussian elimination transforms any matrix into its reduced row echelon form.
\end{theorem}
\begin{definition}
A solution to a system of equations of the form
\[s = s_0 + t_1u_1 + t_2u_2 + \dots +t_{n-r}u_{n-r},\]
where $r$ is the number of nonzero solutions in $A'$ ($r \leq m$), is called a \textbf{general solution} of the system $Ax = b$. It expresses an arbitrary solution $s$ of $Ax = b$ in terms of $n - r$ parameters.
\end{definition}
\begin{theorem}
\hfill\\
Let $Ax = b$ be a system of $r$ nonzero equations in $n$ unknowns. Suppose that $\rank{A} = \rank{A|b}$ and that $(A|b)$ is in reduced row echelon form. Then
\begin{enumerate}
\item $\rank{A} = r$.
\item If the general solution obtained by the procedure above is of the form
\[s = s_0 + t_1u_1 + t_2u_2 + \dots + t_{n-r}u_{n-r},\]
then $\{u_1, u_2, \dots, u_{n-r}\}$ is a basis for the solution set of the corresponding homogeneous system, and $s_0$ is a solution to the original system.
\end{enumerate}
\end{theorem}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$, where $r > 0$, and let $B$ be the reduced row echelon form of $A$. Then
\begin{enumerate}
\item The number of nonzero rows in $B$ is $r$.
\item For each $i = 1, 2, \dots, r$, there is a column $b_{j_i}$ of $B$ such that $b_{j_i} = e_i$.
\item The columns of $A$ numbered $j_1, j_2, \dots, j_r$ are linearly independent.
\item For each $k = 1, 2, \dots, n$, if column $k$ of $B$ is $d_1e_1+d_2e_2+\dots+d_re_r$, then column $k$ of $A$ is $d_1a_{j_1} + d_2a_{j_2} + \dots + d_ra_{j_r}$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
The reduced row echelon form of a matrix is unique.
\end{corollary}
chapter-3/systems-of-linear-equations-theoretical-aspects.tex
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\section{Systems of Linear Equations -- Theoretical Aspects}
\begin{definition}
\hfill\\
The system of equations
\begin{equation}\label{eq:S}
\tag{S}
\begin{split}
a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n = b_1\\
a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n = b_2\\
\dots \\
a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n = b_m,
\end{split}
\end{equation}
where $a_{ij}$ and $b_i$ ($1 \leq i \leq m$ and $1 \leq j \leq n$) are scalars in a field $\F$ and $x_1, x_2, \dots, x_n$ are $n$ variables taking values in $\F$, is a called a \textbf{system of $m$ linear equations in $n$ unknowns over the field $\F$}.
The $m \times n$ matrix
\[\begin{pmatrix}
a_{11} & a_{12} & \dots & a_{1n} \\
a_{21} & a_{22} & \dots & a_{2n} \\
\vdots & \vdots & & \vdots \\
a_{m1} & a_{m2} & \dots & a_{mn}
\end{pmatrix}\]
is called the \textbf{coefficient matrix} of the system \eqref{eq:S}.
If we let
\[x = \begin{pmatrix}
x_1 \\ x_2 \\ \vdots \\ x_n
\end{pmatrix}\ \ \text{and}\ \ b = \begin{pmatrix}
b_1 \\ b_2 \\ \vdots \\ b_m
\end{pmatrix},\]
then the system \eqref{eq:S} may be rewritten as a single matrix equation
\[Ax = b.\]
To exploit the results that we have developed, we often consider a system of linear equations as a single matrix equation.
A \textbf{solution} to the system \eqref{eq:S} is an $n$-tuple
\[s = \begin{pmatrix}
s_1 \\ s_2 \\ \vdots \\ s_n
\end{pmatrix} \in \F^n\]
such that $As = b$. The set of all solutions to the system \eqref{eq:S} is called the \textbf{solution set} of the system. System \eqref{eq:S} is called \textbf{consistent} if its solution set is nonempty; otherwise it is called \textbf{inconsistent}.
\end{definition}
\begin{definition}
\hfill\\
A system $Ax = b$ of $m$ linear equations in $n$ unknowns is said to be \textbf{homogeneous} if $b = 0$. Otherwise the system is said to be \textbf{nonhomogeneous}.\\
Any homogeneous system has at least one solution, namely, the zero vector.
\end{definition}
\begin{theorem}
\hfill\\
Let $Ax = 0$ be a homogeneous system of $m$ linear equations in $n$ unknowns over a field $\F$. Let $K$ denote the set of all solutions to $Ax = 0$. Then $K = \n{L_A}$; hence $K$ is a subspace of $\F^n$ of dimension $n - \rank{L_A} = n - \rank{A}$.
\end{theorem}
\begin{corollary}
\hfill\\
If $m < n$, the system $Ax = 0$ has a nonzero solution.
\end{corollary}
\begin{definition}
\hfill\\
We refer to the equation $Ax = 0$ as the \textbf{homogeneous system corresponding to $Ax = b$}.
\end{definition}
\begin{theorem}
\hfill\\
Let $K$ be the solution set of a system of linear equations $Ax = b$, and let $\mathsf{K}_\mathsf{H}$ be the solution set of the corresponding homogeneous system $Ax = 0$. Then for any solution $s$ to $Ax = b$
\[K = \{s\} + \mathsf{K}_\mathsf{H} = \{s + k: k \in \mathsf{K}_\mathsf{H}\}.\]
\end{theorem}
\begin{theorem}
\hfill\\
Let $Ax = b$ be a system of $n$ linear equations in $n$ unknowns. If $A$ is invertible, then the system has exactly one solution, namely, $A^{-1}b$. Conversely, if the system has exactly one solution, then $A$ is invertible.
\end{theorem}
\begin{definition}
\hfill\\
The matrix $(A|b)$ is called the \textbf{augmented matrix of the system $Ax = b$}.
\end{definition}
\begin{theorem}
\hfill\\
Let $Ax = b$ be a system of linear equations. Then the system is consistent if and only if $\rank{A} = \rank{A|b}$.
\end{theorem}
\begin{definition}
Consider a system of linear equations
\[\begin{split}
a_{11}p_1 + a_{12}p_2 + \dots + a_{1m}p_m = p_1 \\
a_{21}p_1 + a_{22}p_2 + \dots + a_{2m}p_m = p_2 \\
\dots \\
a_{n1}p_1 + a_{n2}p_2 + \dots + a_{nm}p_m = p_m \\
\end{split}\]
This system can be written as $Ap = p$, where
\[p = \begin{pmatrix}
p_1 \\ p_2 \\ \vdots \\ p_m
\end{pmatrix}\]
and $A$ is the coefficient matrix of the system. In this context, $A$ is called the \textbf{input-ouput (or consumption) matrix}, and $Ap = p$ is called the \textbf{equilibrium condition}.
For vectors $b = (b_1, b_2, \dots, b_n)$ and $c = (c_1, c_2, \dots, c_n)$ in $\R^n$, we use the notation $b \geq c$ [$b > c$] to mean $b_i \geq c_i$ [$b_i > c_i$] for all $i$. The vector $b$ is called \textbf{nonnegative [positive]} if $b \geq 0$ [$b > 0$].
\end{definition}
\begin{theorem}
\hfill\\
Let $A$ be an $n \times n$ input-output matrix having the form
\[A = \begin{pmatrix}
B & C \\
D & E
\end{pmatrix},\]
where $D$ is a $1 \times (n -1)$ positive vector and $C$ is an $(n-1)\times 1$ positive vector. Then $(I -A)x = 0$ has a one-dimensional solution set that is generated by a nonnegative vector.
\end{theorem}
chapter-3/the-rank-of-a-matrix-and-matrix-inverses.tex
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\section{The Rank of a Matrix and Matrix Inverses}
\begin{definition}
\hfill\\
If $A \in M_{m \times n}(\F)$, we define the \textbf{rank} of $A$, denoted $\rank{A}$, to be the rank of the linear transformation $L_A: \F^n \to \F^m$.
\end{definition}
\begin{theorem}
\hfill\\
Let $T: V \to W$ be a linear transformation between finite-dimensional vector spaces, and let $\beta$ and $\gamma$ be ordered bases for $V$ and $W$, respectively. Then $\rank{T} = \rank{[T]_\beta^\gamma}$.
\end{theorem}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix. if $P$ and $Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively, then
\begin{enumerate}
\item $\rank{AQ} = \rank{A}$,
\item $\rank{PA} = \rank{A}$,\\ and therefore
\item $\rank{PAQ} = \rank{A}$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Elementary row and column operations on a matrix are rank preserving.
\end{corollary}
\begin{theorem}
\hfill\\
The rank of any matrix equals the maximum number of its linearly independent columns; that is, the rank of a matrix is the dimension of the subspace generated by its columns.
\end{theorem}
\begin{theorem}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$. Then $r \leq m$, $r \leq n$, and, by means of a finite number of elementary row and column operations, $A$ can be transformed into the matrix
\[D = \begin{pmatrix}
I_r & O_1 \\
O_2 & O_3
\end{pmatrix}\]
where $O_1$, $O_2$ and $O_3$ are the zero matrices. Thus $D_{ii} = 1$ for $i \leq r$ and $D_{ij} = 0$ otherwise.
\end{theorem}
\begin{corollary}
\hfill\\
Let $A$ be an $m \times n$ matrix of rank $r$. Then there exist invertible matrices $B$ and $C$ of sizes $m \times m$ and $n \times n$, respectively, such that $D=BAC$, where
\[D = \begin{pmatrix}
I_r & O_1 \\
O_2 & O_3
\end{pmatrix}\]
is the $m \times n$ matrix in which $O_1$, $O_2$, and $O_3$ are zero matrices.
\end{corollary}
\begin{corollary}
\hfill\\
Let $A$ be an $m \times n$ matrix. Then
\begin{enumerate}
\item $\rank{A^t} = \rank{A}$.
\item The rank of any matrix equals the maximum number of its linearly independent rows; that is, the rank of a matrix is the dimension of the subspace generated by its rows.
\item The rows and columns of any matrix generate subspaces of the same dimension, numerically equal to the rank of the matrix.
\end{enumerate}
\end{corollary}
\begin{corollary}
\hfill\\
Every invertible matrix is a product of elementary matrices.
\end{corollary}
\begin{theorem}
\hfill\\
Let $T: V \to W$ and $U: W \to Z$ be linear transformations on finite-dimensional vector spaces $V$, $W$, and $Z$, and let $A$ and $B$ be matrices such that the product $AB$ is defined. Then
\begin{enumerate}
\item $\rank{UT} \leq \rank{U}$.
\item $\rank{UT} \leq \rank{T}$.
\item $\rank{AB} \leq \rank{A}$.
\item $\rank{AB} \leq \rank{B}$.
\end{enumerate}
\end{theorem}
\begin{definition}
\hfill\\
Let $A$ and $B$ be $m \times n$ and $m \times p$ matrices, respectively. By the \textbf{augmented matrix} $(A|B)$, we mean the $m \times (n \times p)$ matrix $(A\ B)$, that is, the matrix whose first $n$ columns are the columns of $A$, and whose last $p$ columns are the columns of $B$.
\end{definition}
chapter-4/a-characterization-of-the-determinant.tex
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\section{A Characterization of the Determinant}
\begin{definition}
\hfill\\
A function $\delta: M_{n \times n}(\F) \to \F$ is called an \textbf{\textit{n}-linear function} if it is a linear function of each row of an $n \times n$ matrix when the remaining $n-1$ rows are held fixed, that is, $\delta$ is $n$-linear if, for every $r = 1, 2, \dots, n$, we have
\[\delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u+kv \\ a_{r + 1} \\ \vdots \\ a_n
\end{pmatrix} = \delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r + 1} \\ \vdots \\ a_n
\end{pmatrix} + k\delta\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}\]
whenever $k$ is a scalar and $u,v$ and each $a_i$ are vectors in $\F^n$.
\end{definition}
\begin{definition}
\hfill\\
An $n$-linear function $\delta: M_{n \times n}(\F) \to \F$ is called \textbf{alternating} if, for each $A \ in M_{n \times n}(\F)$, we have $\delta(A) = 0$ whenever two adjacent rows of $A$ are identical.
\end{definition}
\begin{theorem}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function.
\begin{enumerate}
\item If $A \in M_{n \times n}(\F)$ and $B$ is a matrix obtained from $A$ by interchanging any two rows of $A$, then $\delta(B) = -\delta(A)$.
\item If $A \in M_{n \times n}(\F)$ has two identical rows, then $\delta(A) = 0$.
\end{enumerate}
\end{theorem}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function. If $B$ is a matrix obtained from $A \in M_{n \times n}(\F)$ by adding a multiple of some row of $A$ to another row, then $\delta(B) = \delta(A)$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function. if $M \in M_{n \times n}(\F)$ has rank less than $n$, then $\delta(M) = 0$.
\end{corollary}
\begin{corollary}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function, and let $E_1, E_2$ and $E_3$ in $M_{n \times n}(\F)$ be elementary matrices of types 1, 2, and 3, respectively. Suppose that $E_2$ is obtained by multiplying some row of $I$ by the nonzero scalar $k$. Then $\delta(E_1) = -\delta(I)$, $\delta(E_2) = k \cdot \delta(I)$, and $\delta(E_3) = \delta(I)$.
\end{corollary}
\begin{theorem}
\hfill\\
Let $\delta: M_{n \times n}(\F) \to \F$ be an alternating $n$-linear function such that $\delta(I) = 1$. For any $A,B \in M_{n \times n}(\F)$, we have $\delta(AB) = \delta(A) \cdot \delta(B)$.
\end{theorem}
\begin{theorem}
\hfill\\
If $\delta: M_{n \times n}(\F) \to \F$ is an alternating $n$-linear function such that $\delta(I) = 1$, then $\delta(A) = \det(A)$ for every $A \in M_{n \times n}(\F)$.
\end{theorem}
chapter-4/determinants-of-order-2.tex
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\section{Determinants of Order 2}
\begin{definition}
\hfill\\
If
\[A = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}\]
is a $2 \times 2$ matrix with entries from a field $\F$, then we define the \textbf{determinant} of $A$, denoted $\det(A)$ or $|A|$, to be the scalar $ad-bc$.
\end{definition}
\begin{theorem}
\hfill\\
The function $\det: M_{2 \times 2}(\F) \to \F$ is a linear function of each row of a $2 \times 2$ matrix when the other row is held fixed. That is, if $u$, $v$ and $w$ are in $\F^2$ and $k$ is a scalar, then
\[\det \begin{pmatrix}
u + kv \\
w
\end{pmatrix} = \det\begin{pmatrix}
u \\ w
\end{pmatrix} + k\det\begin{pmatrix}
v \\ w
\end{pmatrix}\]
and
\[\det\begin{pmatrix}
w \\ u + kv
\end{pmatrix} = \det\begin{pmatrix}
w \\ u
\end{pmatrix} + k \det \begin{pmatrix}
w \\ v
\end{pmatrix}.\]
\end{theorem}
\begin{theorem}\label{Theorem 4.2}
\hfill\\
Let $A \in M_{2 \times 2}(\F)$. Then the determinant of $A$ is nonzero if and only if $A$ is invertible. Moreover, if $A$ is invertible, then
\[A^{-1} = \frac{1}{\det(A)}\begin{pmatrix}
A_{22} & -A_{12} \\
-A_{21} & A_{11}
\end{pmatrix}.\]
\end{theorem}
\begin{definition}
\hfill\\
By the \textbf{angle} between two vectors in $\R^2$, we mean the angle with measure $\theta$ ($0 \leq \theta < \pi$) that is formed by the vectors having the same magnitude and direction as the given vectors by emanating from the origin.
\end{definition}
\begin{definition}
\hfill\\
If $\beta = \{u,v\}$ is an ordered basis for $\R^2$, we define the \textbf{orientation} of $\beta$ to be the real number
\[O\begin{pmatrix}
u \\ v
\end{pmatrix} = \frac{\det\begin{pmatrix}
u \\ v
\end{pmatrix}}{\abs{\det\begin{pmatrix}
u \\ v
\end{pmatrix}}}\]
(The denominator of this fraction is nonzero by \autoref{Theorem 4.2}).
\end{definition}
\begin{definition}
\hfill\\
A coordinate system $\{u, v\}$ is called \textbf{right-handed} if $u$ can be rotated in a counterclockwise direction through an angle $\theta$ ($0 < \theta < \pi$) to coincide with $v$. Otherwise, $\{u ,v\}$ is called a \textbf{left-handed} system.
\end{definition}
\begin{definition}
\hfill\\
Any ordered set $\{u, v\}$ in $\R^2$ determines a parallelogram in the following manner. Regarding $u$ and $v$ as arrows emanating from the origin of $\R^2$, we call the parallelogram having $u$ and $v$ as adjacent sides the \textbf{parallelogram determined by $u$ and $v$}.
\end{definition}
chapter-4/determinants-of-order-n.tex
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\section{Determinants of Order $n$}
\section{Determinants of Order \textit{n}}
\begin{notation}
\hfill\\
Given $A \in M_{n \times n}(\F)$, for $n \geq 2$, denote the $(n-1) \times (n - 1)$ matrix obtained from $A$ by deleting row $i$ and column $j$ by $\tilde{A}_{ij}$. Thus for
\[A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix} \in M_{3 \times 3}(\R)\]
we have
\[\tilde{A}_{11} = \begin{pmatrix}
5 & 6 \\
8 & 9
\end{pmatrix},\ \ \ \ \
\tilde{A}_{13}=\begin{pmatrix}
4 & 5 \\
7 & 8
\end{pmatrix},\ \ \ \ \
\tilde{A}_{32} = \begin{pmatrix}
1 & 3 \\
4 & 6
\end{pmatrix}\]
and for
\[B = \begin{pmatrix}
1 & -1 & 2 & -1 \\
-3 & 4 & 1 & -1 \\
2 & -5 & -3 & 8 \\
-2 & 6 & -4 & 1
\end{pmatrix}\]
we have
\[\tilde{B}_{23} = \begin{pmatrix}
1 & -1 & -1 \\
2 & -5 & 8 \\
-2 & 6 & 1
\end{pmatrix}\ \ \ \ \ \text{and}\ \ \ \ \ \tilde{B}_{42}=\begin{pmatrix}
1 & 2 & -1 \\
-3 & 1 & -1 \\
2 & -3 & 8
\end{pmatrix}\]
\end{notation}
\begin{definition}
\hfill\\
Let $A \in M_{n \times n}(\F)$. If $n =1$, so that $A = (A_{11})$, we define $\det(A) = A_{11}$. For $n \geq 2$, we define $\det(A)$ recursively as
\[\det(A) = \sum_{j=1}^{n}(-1)^{1+j}A_{1j}\cdot\det(\tilde{A}_{1j}).\]
The scalar $\det(A)$ is called the \textbf{determinant} of $A$ and is also denoted by $|A|$. The scalar
\[(-1)^{i+j}\det(\tilde{A}_{ij})\]
is called the \textbf{cofactor} of the entry of $A$ in row $i$, column $j$.
\end{definition}
\begin{definition}
\hfill\\
Letting
\[c_{ij} = (-1)^{i+j}\det(\tilde{A}_{ij})\]
denote the cofactor of the row $i$, column $j$ entry of $A$, we can express the formula for the determinant of $A$ as
\[\det(A) = A_{11}c_{11} + A_{12}c_{12}+\dots+A_{1n}c_{1n}.\]
Thus the determinant of $A$ equals the sum of the products of each entry in row $1$ of $A$ multiplied by its cofactor. This formula is called \textbf{cofactor expansion along the first row} of $A$.
\end{definition}
\begin{theorem}
\hfill\\
the determinant of an $n \times n$ matrix is a linear function of each row when the remaining rows are held fixed. That is, for $1 \leq r \leq n$, we have
\[\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u+kv \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}=\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ u \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix} + k\det\begin{pmatrix}
a_1 \\ \vdots \\ a_{r-1} \\ v \\ a_{r+1} \\ \vdots \\ a_n
\end{pmatrix}\]
wherever $k$ is a scalar and $u, v$ and each $a_i$ are row vectors in $\F^n$.
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has a row consisting entirely of zeros, then $\det(A) = 0$.
\end{corollary}
\begin{lemma}
\hfill\\
Let $B \in M_{n \times n}(\F)$, where $n \geq 2$. If row $i$ of $B$ equals $e_k$ for some $k$ ($1 \leq k \leq n$), then $\det(B) = (-1)^{i+k}\det(\tilde{B}_{ik})$.
\end{lemma}
\begin{theorem}
\hfill\\
The determinant of a square matrix can be evaluated by cofactor expansion along any row. That is, if $A \in M_{n \times n}(\F)$, then for any integer $i$ ($1 \leq i \leq n$),
\[\det(A) = \sum_{j=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij}).\]
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has two identical rows, then $\det(A) = 0$.
\end{corollary}
\begin{theorem}
\hfill\\
If $A \in M_{n \times n}(\F)$ and $B$ is a matrix obtained from $A$ by interchanging any two rows of $A$, then $\det(B) = -\det(A)$.
\end{theorem}
\begin{theorem}
\hfill\\
Let $A \in M_{n \times n}(\F)$, and let $B$ be a matrix obtained by adding a multiple of one row of $A$ to another row of $A$. Then $\det(B) = \det(A)$.
\end{theorem}
\begin{corollary}
\hfill\\
If $A \in M_{n \times n}(\F)$ has rank less than $n$, then $\det(A) = 0$.
\end{corollary}
\begin{remark}\label{Remark 4.1}
\hfill\\
The following rules summarize the effect of an elementary row operation on the determinant of a matrix $A \ in M_{n \times n}(\F)$.
\begin{enumerate}
\item If $B$ is a matrix obtained by interchanging any two rows of $A$, then $\det(B) = -\det(A)$.
\item If $B$ is a matrix obtained by multiplying a row of $A$ by a nonzero scalar $k$, then $\det(B) = k\det(A)$.
\item If $B$ is a matrix obtained by adding a multiple of one row of $A$ to another row of $A$, then $\det(B) = \det(A)$.
\end{enumerate}
\end{remark}
\begin{lemma}
\hfill\\
The determinant of an upper triangular matrix is the product of its diagonal entries.
\end{lemma}
chapter-4/properties-of-determinants.tex
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\section{Properties of Determinants}
\begin{remark}
\hfill\\
Because the determinant of the $n \times n$ matrix is $1$, we can interpret \autoref{Remark 4.1} as the following facts about the determinants of elementary matrices.
\begin{enumerate}
\item If $E$ is an elementary matrix obtained by interchanging any two rows of $I$, then $\det(E) = -1$.
\item If $E$ is an elementary matrix obtained by multiplying some row of $I$ by the nonzero scalar $k$, then $\det(E) = k$.
\item If $E$ is an elementary matrix obtained by adding a multiple of some row of $I$ to another row, then $\det(E) = 1$.
\end{enumerate}
\end{remark}
\begin{theorem}
\hfill\\
For any $A, B \in M_{n \times n}(\F)$, $\det(AB) = \det(A) \cdot \det(B)$.
\end{theorem}
\begin{corollary}
\hfill\\
A matrix $A \in M_{n \times n}(\F)$ is invertible if and only if $\det(A) \neq 0$. Furthermore, if $A$ is invertible, then $\det(A^{-1}) = \displaystyle\frac{1}{\det(A)}$.
\end{corollary}
\begin{theorem}
\hfill\\
For any $A \in M_{n \times n}(\F)$, $\det(A^t)=\det(A)$.
\end{theorem}
\begin{theorem}[\textbf{Cramer's Rule}]
\hfill\\
Let $Ax = b$ be the matrix form of a system of $n$ linear equations in $n$ unknowns, where $x = (x_1, x_2, \dots, x_n)^t$. If $\det(A) \neq 0$, then this system has a unique solution, and for each $k$ ($k = 1, 2, \dots, n$),
\[x_k = \frac{\det(M_k)}{\det(A)},\]
where $M_k$ is the $n \times n$ matrix obtained from $A$ by replacing column $k$ of $A$ by $b$.
\end{theorem}
\begin{definition}
\hfill\\
It is possible to interpret the determinant of a matrix $A \in M_{n \times n}(\R)$ geometrically. If the rows of $A$ are $a_1, a_2, \dots, a_n$, respectively, then $|\det(A)|$ is the \textbf{\textit{n}-dimensional volume} (the generalization of are in $\R^2$ and volume in $\R^3$) of the parallelepiped having the vectors $a_1, a_2, \dots, a_n$ as adjacent sides.
\end{definition}
\begin{definition}
\hfill\\
A matrix $M \in M_{n \times n}(\C)$ is called \textbf{nilpotent} if, for some positive integer $k$, $M^k = O$, where $O$ is the $n \times n$ zero matrix.
\end{definition}
\begin{definition}
\hfill\\
A matrix $M \in M_{n \times n}(\C)$ is called \textbf{skew-symmetric} if $M^t = -M$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $Q \in M_{n \times n}(\R)$ is called \textbf{orthogonal} if $QQ^t = I$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $Q \in M_{n \times n}(\C)$ is called \textbf{unitary} if $QQ^* = I$, where $Q^* = \overline{Q^t}$.
\end{definition}
\begin{definition}
\hfill\\
A matrix $A \in M_{n \times n}(\F)$ is called \textbf{lower triangular} if $A_{ij}=0$ for $1 \leq i < j \leq n$.
\end{definition}
\begin{definition}
\hfill\\
A matrix of the form
\[\begin{pmatrix}
1 & c_0 & c_0^2 & \dots & c_0^n \\
1 & c_1 & c_1^2 & \dots & c_1^n \\
\vdots & \vdots & \vdots & &\vdots \\
1 & c_n & c_n^2 & \dots & c_n^n
\end{pmatrix}\]
is called a \textbf{Vandermonde matrix}.
\end{definition}
\begin{definition}
\hfill\\
Let $A \in M_{n \times n}(\F)$ be nonzero. For any $m$ ($1 \leq m \leq n$), and $m \times m$ \textbf{submatrix} is obtained by deleting any $n - m$ rows and any $n - m$ columns of $A$.
\end{definition}
\begin{definition}
\hfill\\
The \textbf{classical adjoint} of a square matrix $A$ is the transpose of the matrix whose $ij$-entry is the $ij$-cofactor of $A$.
\end{definition}
\begin{definition}
\hfill\\
Let $y_1, y_2, \dots, y_n$ be linearly independent function in $\C^\infty$. For each $y \in \C^\infty$, define $T(y) \in \C^\infty$ by
\[[T(y)](t) = \det\begin{pmatrix}
y(t) & y_1(t) & y_2(t) & \dots & y_n(t) \\
y'(t) & y'_1(t) & y'_2(t) & \dots & y'_n(t) \\
\vdots & \vdots & \vdots & &\vdots \\
y^{(n)}(t) & y_1^{(n)}(t) & y_2^{(n)}(t) & \dots & y_n^{(n)}(t)
\end{pmatrix}\]
The preceding determinant is called the \textbf{Wronskian} of $y, y_1, \dots, y_n$.
\end{definition}
chapter-4/summary-important-facts-about-determinants.tex
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\section{Summary -- Important Facts about Determinants}
\begin{definition}
\hfill\\
The \textbf{determinant} of an $n \times n$ matrix $A$ having entries from a field $\F$ is a scalar in $\F$, denoted by $\det(A)$ or $|A|$, and can be computed in the following manner:
\begin{enumerate}
\item If $A$ is $1 \times 1$, then $\det(A) = A_{11}$, the single entry of $A$.
\item If $A$ is $2 \times 2$, then $\det(A) = A_{11}A_{22} - A_{12}A_{21}$.
\item If $A$ is $n \times n$ for $n > 2$, then
\[\det(A) = \sum_{j=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]
(if the determinant is evaluated by the entries of row $i$ of $A$) or
\[\det(A) = \sum_{i=1}^{n}(-1)^{i+j}A_{ij}\cdot\det(\tilde{A}_{ij})\]
(if the determinant is evaluated by the entries of column $j$ of $A$), where $\tilde{A}_{ij}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row $i$ and column $j$ from $A$.
\end{enumerate}
In the formulas above, the scalar $(-1)^{i+j}\det(\tilde{A}_{ij})$ is called the \textbf{cofactor} of the row $i$ column $j$ of $A$.
\end{definition}
\begin{definition}[\textbf{Properties of the Determinant}]
\begin{enumerate}
\item[]
\item If $B$ is a matrix obtained by interchanging any two rows or interchanging any two columns of an $n \times n$ matrix $A$, then $\det(B) = -\det(A)$.
\item If $B$ is a matrix obtained by multiplying each entry of some row or column of an $n \times n$ matrix $A$ by a scalar $k$, then $\det(B) = k\cdot\det(A)$.
\item If $B$ is a matrix obtained from an $n \times n$ matrix $A$ by adding a multiple of row $i$ to row $j$ or a multiple of column $i$ to column $j$ for $i \neq j$, then $\det(B) = \det(A)$.
\end{enumerate}
\end{definition}
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\newcommand{\F}{\mathbb{F}}
% Theorem Styles
\declaretheorem[numberwithin=section, style=definition]{theorem, definition, notation, lemma, corollary, remark, example}
\declaretheorem[numberwithin=chapter, style=definition]{theorem, definition, notation, lemma, corollary, remark, example}
% Formatting
\setlist[enumerate]{font=\bfseries}