89 lines
3.2 KiB
TeX
89 lines
3.2 KiB
TeX
\section{The Rank of a Matrix and Matrix Inverses}
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\begin{definition}
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\hfill\\
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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 $L_A: \F^n \to \F^m$.
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\end{definition}
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\begin{theorem}
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\hfill\\
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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 $\rank{T} = \rank{[T]_\beta^\gamma}$.
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\end{theorem}
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\begin{theorem}
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\hfill\\
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Let $A$ be an $m \times n$ matrix. if $P$ and $Q$ are invertible $m \times m$ and $n \times n$ matrices, respectively, then
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\begin{enumerate}
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\item $\rank{AQ} = \rank{A}$,
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\item $\rank{PA} = \rank{A}$,\\ and therefore
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\item $\rank{PAQ} = \rank{A}$.
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\end{enumerate}
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\end{theorem}
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\begin{corollary}
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\hfill\\
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Elementary row and column operations on a matrix are rank preserving.
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\end{corollary}
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\begin{theorem}
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\hfill\\
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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.
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\end{theorem}
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\begin{theorem}
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\hfill\\
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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
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\[D = \begin{pmatrix}
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I_r & O_1 \\
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O_2 & O_3
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\end{pmatrix}\]
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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.
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\end{theorem}
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\begin{corollary}
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\hfill\\
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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
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\[D = \begin{pmatrix}
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I_r & O_1 \\
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O_2 & O_3
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\end{pmatrix}\]
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is the $m \times n$ matrix in which $O_1$, $O_2$, and $O_3$ are zero matrices.
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\end{corollary}
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\begin{corollary}
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\hfill\\
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Let $A$ be an $m \times n$ matrix. Then
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\begin{enumerate}
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\item $\rank{A^t} = \rank{A}$.
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\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.
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\item The rows and columns of any matrix generate subspaces of the same dimension, numerically equal to the rank of the matrix.
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\end{enumerate}
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\end{corollary}
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\begin{corollary}
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\hfill\\
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Every invertible matrix is a product of elementary matrices.
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\end{corollary}
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\begin{theorem}
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\hfill\\
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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
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\begin{enumerate}
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\item $\rank{UT} \leq \rank{U}$.
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\item $\rank{UT} \leq \rank{T}$.
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\item $\rank{AB} \leq \rank{A}$.
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\item $\rank{AB} \leq \rank{B}$.
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\end{enumerate}
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\end{theorem}
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\begin{definition}
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\hfill\\
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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 $B$.
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\end{definition}
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