How do I find transformation matrix with respect to given basis in the domain and/or the codomain, given the transformation in the standard basis?

Question

I´m being given a linear transformation, for which I can find the standard basis in the domain and codomain; but then, the book ask to find the associated matrix related to a new basis for the domain... then the same for a new basis in the codomain... then for both together. There´s an example showing the mechanics but I can´t get why it´s the way it is; I just see they multiply or pre-multiply the matrix in the standard basis with the new basis. Could you please help me understand why this solves the question?

Edit:

Example:

Given $T: \mathbb{R}^2 \to\mathbb{R}^3$ defined by the following matrix in standard basis:

$$T(\mathbf{x})=\begin{bmatrix}1&1\\-1&0\\0&0\end{bmatrix}\mathbf{x}$$

Find the associated matrix being the domain basis: $\{(1,3)^\intercal,(-2,4)^\intercal\}$. Same but being the codomain basis: $\{(1,1,1)^\intercal,(2,2,0)^\intercal,(3,0,0)^\intercal\}$. Then, both together.

Thanks!!

Answer 1

The General Matrix Representation Theorem:

Let $T:V\to W$ be a linear map from a n-dimensional vector space V to an m-dimensional vector space W, and let $B_W = \{\vec w_1,...,\vec w_m\}$ be an ordered basis for W. Then, there is a unique m × n matrix A such that $[T(\vec v)]_{B_W} = A[\vec v]_{B_V}.$

From this, we have an algorithm for constructing a matrix representation for a linear map:

1. Find a basis $B_V = \{\vec v_1,...,\vec v_n\}$ for the domain V and a basis $B_V = \{\vec w_1,...,\vec w_m\}$ for the codomain W.
2. Find the function values $T(\vec v_j), 1\le j\le n$, of the domain basis vectors.
3. Find the coordinate vectors $[T(\vec v)]_{B_W}$ of the function values $T(\vec v_j)$ with respect to the codomain basis $B_W$.
4. Construct the m × n matrix A with the coordinate vectors $[T(\vec v_j)]_{B_W}, 1\le j\le n$, as its columns.