Transformation Matrix

A way of representing Linear Map as a matrix.

Theorem

Let $V,W$ be finite dimensional vector spaces with $dim(V)=n$ and $dim(W)=k$. Fix bases $\alpha =\{v_1,\dots ,v_n\}$ and $\beta =\{w_1,\dots ,w_k\}$ With $V=span(\alpha )$, $W=span(\beta )$

Suppose that $T:V\to W$ is a linear transformation with: $T(v_i)=a_{1i}{w}_1+\cdots +a_{ki}{w}_k={\sum }_{j=1}^k{a}_{ji}{w}_j$ The matrix is represented as:

a_{11} \dots a_{1n} \\ \vdots\\ a_{k1} \dots a_{kn} \end{array}\right]

It starts at basis $\alpha$ and ends in basis $\beta$

Proof

• Let $A=(A_{ij})\in {\mathbb{F}}^{m\times n}$ where scalars $A_{ij}$ are obtained by $T_{\alpha j}={\sum }_{i=1}^m{A}_{ij}{B}_i,\forall 1\le j\le n$ where $\beta =\{{\alpha }_1,\dots ,a_n\}$, ${\beta }^{\prime }=\{{\beta }_1,\dots ,{\beta }_m\}$
• i.e, $$[T_{\alpha i}]{\beta’} = \left[\begin{array}{c} A{1j} \ A_{2j} \ \vdots\ A_{mj} \ \end{array}\right]

- Now, let $\alpha \in V$, Then $\alpha = c_{1}a_{1} +\dots+c_{n}a_{n}$ - Thus, $T(a) = T\left( \sum_{n}^{j=1}c_{j}\alpha_{j} \right) = \sum_{n}^{j=1}(\sum_{m}^{i=1}c_{j}A_{ij})B_{i}$ - Thus, $$[T_{\alpha i}]_{\beta'} = \left[\begin{array}{c} \sum_{n}^{j=1}c_{j}A_{1j} \\ \sum_{n}^{j=1}c_{j}A_{2j} \\ \vdots\\ \sum_{n}^{j=1}c_{j}A_{mj} \\ \end{array}\right]

• Notice that, $$A[\alpha]{\beta} = \left[\begin{array}{c} \sum{n}^{j=1}c_{j}A_{1j} \ \sum_{n}^{j=1}c_{j}A_{2j} \ \vdots\ \sum_{n}^{j=1}c_{j}A_{mj} \ \end{array}\right]

- Thus, a matrix exists - This matrix is unique, as the linear transformation is unique # Examples - [[Finding Matrix Representation for Identity Transformation]]