[Math] Let T be a linear transformation, find T

linear algebralinear-transformations

Ok, I was asked this strange question that I can't seem to grasp the concept of..

Let $T$ be a linear transformation such that:
$$T \langle1,-1\rangle = \langle 0,3\rangle \\
T \langle2, 3\rangle = \langle 5,1\rangle $$
Find $T$.

Is there suppose to be a function out of this? A matrix of some kind? Maybe both? If so, what is it?

Best Answer

Yes, $T$ is a matrix of some kind. You can tell that it is $2\times 2$ since both its inputs and outputs are vectors of length $2$. My recommendation for solving this is the following. Suppose $T$ has matrix representation

$$ T \;\; =\;\; \left [ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ]. $$

Start with the first equation which you can write as

$$ \left [ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right ] \left [ \begin{array}{c} 1 \\ -1\\ \end{array} \right ] \;\; =\;\; \left [ \begin{array}{c} 0 \\ 3\\ \end{array} \right ]. $$

You can find the values of $a,b,c,d$ by working through these equations.