Matrix Representation of Linear Maps

Henceforth we fix an ordering on a basis \(\beta=\{v_1,...,v_n\}\)

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: \(\beta=\{v_i\}\) of \(V\) and \(\gamma=\{w_i\}\) 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 \(\alpha=\{v_1,...,v_n\},\beta=\{w_1,...,w_m\},\gamma=\{z_1,...,z_p\}\)

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

Thus \(UT:V\rightarrow Z\) is a linear map

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

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