Find matrix representation based on two vector transformations

Question

Find the matrix representation of the linear transformation T that maps the input vector $\vec{x}_1 = (1,1)^T$ to the output vector $\vec{y}_1 = (2,-3)^T$ and maps the input vector $\vec{x}_2 = (1,2)^T$ to the output vector $\vec{y}_2 = (5,1)^T$

I am given a hint that this is a "non-standard basis probing" question.

Independently I see the transformations seem to be $T_1 = (1,-4)$ and $T_2 = (4,-1)$

However I can't seem to create the correct matrix based on those two independent transformations. Any hints?

Answer 1

Hint: suppose the matrix is $V$, then we have $$ \begin{bmatrix} x_1& x_2\\ \end{bmatrix} V = \begin{bmatrix} 1 & 1\\ 1 & 2\\ \end{bmatrix} V = \begin{bmatrix} 2 & 5\\ -3 & 1\\ \end{bmatrix} = \begin{bmatrix} y_1& y_2\\ \end{bmatrix} $$

Consider to find the inverse matrix of $\begin{bmatrix} x_1& x_2\\ \end{bmatrix}$

Answer 2

You can write this as a matrix equation. Create a matrix $X$ from $\vec x_1$ and $\vec x _2$,and another one $Y$ from $\vec y_1$ and $\vec y_2$. Then $TX=Y$. Multiply both sides on the right with $X^{-1}$ and you get the answer. $$T=\begin{pmatrix} 2 & 5 \\ -3 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}^{-1}$$

Answer 3

You know the matrix representation of $T$ under the basis $(1, 1)^T$ and $(1,2)^T$.

Use a change of basis matrix. If $A$ sends $(1,1)^T$ to $(1,0)^T$, and $M$ sends $(1,0)^T$ to $(2,-3)^T$, then their composition sends $(1,1)^T$ to $(2,-3)^T$, and similar for the other basis vector

M is easy to calculate - you know what it sends the standard basis to. $A$ is a bit tricky - it sends $(1,1)^T$ to $(1,0)^T$ and $(1,2)^T$ to $(0,1)^T$.

You know what $A^{-1}$ sends the standard basis to. You can then construct the matrix for $A^{-1}$, invert it to get the matrix for $A$, and then multiply it with $M$. (in the order corresponding to doing A first, then M)