Finding Linear Transformation Matrix $\mathbb{R}^2 \rightarrow\mathbb{R}^2$ and $\mathbb{R}^3 \rightarrow\mathbb{R}^2$

Question

a) Let $T: \mathbb{R}^2 \rightarrow\mathbb{R}^2$ be a linear transformation such that $T(1,1) = (1,2)$ and $T(1,2)=(3,4)$. Find $T(5,8)$ and $T(3,-2)$ ?

b) Let $T: \mathbb{R}^3 \rightarrow\mathbb{R}^2$ be a linear transformation such that $T(1,0,-1) = (2,3)$ and $T(2,1,3) = (-1,0)$. Find $T(8,3,7)$?

Answer 1

Hint: Solve $a(1,1)+b(1,2)=(5,8)$ for $a$ and $b$. You should get $a=2, b=3$.

Now "use linearity". By that I mean $T(5,8)=2T(1,1)+3T(1,2)$.

The other problem is similar.

Answer 2

1. Let the matrix be $$T = \begin{bmatrix} x & z \\ y & w \end{bmatrix} $$

Given $$ T \begin{bmatrix} 1 \\ 1\end{bmatrix} = \begin{bmatrix} 1 \\ 2\end{bmatrix}$$ $$ \Rightarrow \begin{bmatrix} x \\ y\end{bmatrix} + \begin{bmatrix} z \\ w\end{bmatrix} = \begin{bmatrix} 1 \\ 2\end{bmatrix}$$ $$ \Rightarrow x + z = 1, y + w = 2$$ Similarly for second example, $$ x + 2z = 3, y + 2w = 4$$ Which on solving gives, $$T = \begin{bmatrix} -1 & 2 \\ 0 & 2 \end{bmatrix} $$

2. You can't do that analytically. Since there are $3$ unknowns and $2$ equations.