Matrix Representation of Linear Maps

Henceforth we fix an ordering on a basis $\beta=\lbrace v_1,...,v_n\rbrace $

Coordinate vector relative to a basis

Let $x\in V$ then $x=\sum a_iv_i$

Thus, $[x]_\beta=(a_1,...,a_n)$ (the coordinate of $x$ with respect to the basis $\beta$)

Definition

Fix the basis: $\beta=\lbrace v_i\rbrace $ of $V$ and $\gamma=\lbrace w_i\rbrace $ of $W$

Let $T:V\rightarrow W$ linear, $T(v_j)=t_{1j}w_1+...+t_{mj}w_m$

The matrix of $T$ with respect to $\beta$ and $\gamma$ is given by $[T]_\beta^\gamma=[t_{ij}]$ (a $m\times n$ matrix)

$\beta=I^2, \gamma=I^3$, $T(a_1,a_2)=(a_1+3n_2, 0, 2a_1-4a_2)$

Thus, $[T]_\beta^\gamma=$

\[ \begin{bmatrix} 1 & 3 \\ 0 & 0\\ 2 & 4 \end{bmatrix} \]

Let $\mathcal L (V,W)$ be a space of all linear maps from $V$ to $W$

$\mathcal L(V,W)$ is a linear transformation

$\varphi: \mathcal L (V,W)\rightarrow M_{m\times n}(F)$ given by $\varphi(T)=[T]^\gamma_\beta$ is a linear map

For vector space $V,W,Z$, and their basis $\alpha=\lbrace v_1,...,v_n\rbrace ,\beta=\lbrace w_1,...,w_m\rbrace ,\gamma=\lbrace z_1,...,z_p\rbrace $

and $[T]^\beta_\alpha=[t_{ij}]$ $[U]^\gamma_\beta=[u_{ij}]$

Thus $UT:V\rightarrow Z$ is a linear map

$UT(v_j)=U(T(v_j))=U(\sum^k t_{kj}w_k)=\sum^k t_{kj}U(w_k)=\sum^k\sum^l t_{kj}u_{lk}z_l=\sum^l c_{lj}z_l$

$[C]^\gamma_\beta=[UT]^\gamma_\beta=[T]^\beta_\alpha\cdot[U]^\gamma_\beta$