\chapter{List of Symbols}
\begin{align*}
	 & A_{ij}                            & \text{the $ij$-th entry of the matrix $A$}                                       \\
	 & A^{-1}                            & \text{the inverse of the matrix $A$}                                             \\
	 & A^\dag                            & \text{the pseudoinverse of the matrix $A$}                                       \\
	 & A^*                               & \text{the adjoint of the matrix $A$}                                             \\
	 & \tilde{A}_{ij}                    & \text{the matrix $A$ with row $i$ and column $j$ deleted}                        \\
	 & A^t                               & \text{the transpose of the matrix $A$}                                           \\
	 & (A|B)                             & \text{the matrix $A$ augmented by the matrix $B$}                                \\
	 & B_1 \bigoplus \dots \bigoplus B_k & \text{the direct sum of matrices $B_1$ through $B_k$}                            \\
	 & \mathcal{B}(V)                    & \text{the set of bilinear forms on $V$}                                          \\
	 & \beta^*                           & \text{the dual basis of $\beta$}                                                 \\
	 & \beta_x                           & \text{the $T$-cyclic basis generated by $x$}                                     \\
	 & \C                                & \text{the field of complex numbers}                                              \\
	 & \C_i                              & \text{the $i$th Gerschgorin disk}                                                \\
	 & \cond{A}                          & \text{the condition number of the matrix $A$}                                    \\
	 & C^n(\R)                           & \text{set of functions $f$ on $\R$ with $f^{(n)}$ continuous}                    \\
	 & C^\infty                          & \text{set of functions with derivatives of every order}                          \\
	 & C(\R)                             & \text{the vector space of continuous functions on $\R$}                          \\
	 & C([0,1])                          & \text{the vector space of continuous functions on $[0,1]$}                       \\
	 & C_x                               & \text{the $T$-cyclic subspaces generated by $x$}                                 \\
	 & D                                 & \text{the derivative operator on $C^\infty$}                                     \\
	 & \ldet{A}                          & \text{the determinant of the matrix $A$}                                         \\
	 & \delta_{ij}                       & \text{the Kronecker delta}                                                       \\
	 & \ldim{V}                          & \text{the dimension of $V$}                                                      \\
	 & e^A                               & \lim_{m \to \infty} \left(I + A + \frac{A^2}{2!} + \dots + \frac{A^m}{m!}\right) \\
	 & e_i                               & \text{the $i$th standard vector of $\F^n$}                                       \\
\end{align*}
\begin{align*}
	 & E_\lambda              & \text{the eigenspace of $T$ corresponding to $\lambda$}               \\
	 & \F                     & \text{a field}                                                        \\
	 & f(A)                   & \text{the polynomial $f(x)$ evaluated at the matrix $A$}              \\
	 & F^n                    & \text{the set of $n$-tuples with entries in a field $\F$}             \\
	 & f(T)                   & \text{the polynomial $f(x)$ evaluated at the operator $T$}            \\
	 & \mathcal{F}(S,\F)      & \text{the set of functions from $S$ to a field $\F$}                  \\
	 & H                      & \text{space of continuous complex functions on $[0, 2\pi]$}           \\
	 & I_n \text{ or } I      & \text{the $n \times n$ identity matrix}                               \\
	 & \Id_V \text{ or } \Id  & \text{the identity operator on $V$}                                   \\
	 & K_\lambda              & \text{generalized eigenspace of $T$ corresponding to $\lambda$}       \\
	 & K_\phi                 & \{x\ |\ (\phi(T))^p(x) = 0 \text{, for some positive integer $p$}\}   \\
	 & L_A                    & \text{left-multiplication transformation by matrix $A$}               \\
	 & \lim_{m \to \infty}A_m & \text{the limit of a sequence of matrices}                            \\
	 & \linear{V}             & \text{the space of linear transformations from $V$ to $V$}            \\
	 & \linear{V, W}          & \text{the space of linear transformations from $V$ to $W$}            \\
	 & M_{m \times n}(\F)     & \text{the set of $m \times n$ matrices with entries in $\F$}          \\
	 & v(A)                   & \text{the column sum of the matrix $A$}                               \\
	 & v_j(A)                 & \text{the $j$th column sum of the matrix $A$}                         \\
	 & N(T)                   & \text{the null space of $T$}                                          \\
	 & \nullity{T}            & \text{the dimension of the null space of $T$}                         \\
	 & O                      & \text{the zero matrix}                                                \\
	 & \per{M}                & \text{the permanent of the $2 \times 2$ matrix $M$}                   \\
	 & P(\F)                  & \text{the space of polynomials with coefficients in $\F$}             \\
	 & P_n(\F)                & \text{the polynomials in $P(\F)$ of degree at most $n$}               \\
	 & \phi_\beta             & \text{the standard representation with respect to basis $\beta$}      \\
	 & \R                     & \text{the field of real numbers}                                      \\
	 & \rank{A}               & \text{the rank of the matrix $A$}                                     \\
	 & \rank{T}               & \text{the rank of the linear transformation $T$}                      \\
	 & \rho(A)                & \text{the row sum of the matrix $A$}                                  \\
	 & \rho_i(A)              & \text{the $i$th row sum of the matrix $A$}                            \\
	 & R(T)                   & \text{the range of the linear transformation $T$}                     \\
	 & S_1 + S_2              & \text{the sum of sets $S_1$ and $S_2$}                                \\
	 & \lspan{S}              & \text{the span of the set $S$}                                        \\
	 & S^\perp                & \text{the orthogonal complement of the set $S$}                       \\
	 & [T]_\beta              & \text{the matrix representation of $T$ in basis $\beta$}              \\
	 & [T]_\beta^\gamma       & \text{the matrix representation of $T$ in bases $\beta$ and $\gamma$} \\
	 & T^{-1}                 & \text{the inverse of the linear transformation $T$}                   \\
\end{align*}
\begin{align*}
	 & T^\dag                            & \text{the pseudoinverse of the linear transformation $T$} \\
	 & T^*                               & \text{the adjoint of the linear operator $T$}             \\
	 & T_0                               & \text{the zero transformation}                            \\
	 & T^t                               & \text{the transpose of the linear transformation $T$}     \\
	 & T_\theta                          & \text{the rotation transformation by $\theta$}            \\
	 & T_W                               & \text{the restriction of $T$ to a subspace $W$}           \\
	 & \ltr{A}                           & \text{the trace of the matrix $A$}                        \\
	 & V^*                               & \text{the dual space of the vector space $V$}             \\
	 & V/W                               & \text{the quotient space of $V$ modulo $W$}               \\
	 & W_1 + \dots + W_k                 & \text{the sum of subspaces $W_1$ through $W_k$}           \\
	 & \sum_{i=1}^k W_i                  & \text{the sum of subspaces $W_i$ through $W_k$}           \\
	 & W_1 \bigoplus W_2                 & \text{the direct sum of subspaces $W_1$ and $W_2$}        \\
	 & W_1 \bigoplus \dots \bigoplus W_k & \text{the direct sum of subspaces $W_1$ through $W_k$}    \\
	 & \norm{x}                          & \text{the norm of the vector $\vec{x}$}                   \\
	 & [x]_\beta                         & \text{the coordinate vector of $x$ relative to $\beta$}   \\
	 & \langle x, y \rangle              & \text{the inner product of $\vec{x}$ and $\vec{y}$}       \\
	 & \Z_2                              & \text{the field consisting of $0$ and $1$}                \\
	 & \overline{\vec{z}}                & \text{the complex conjugate of $\vec{z}$}                 \\
	 & \vec{0}                           & \text{the zero vector}                                    \\
\end{align*}
