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 $\beta$

Examples

• Finding Matrix Representation for Identity Transformation