I'm stuck on this seemingly simple homework question, but I just don't know how to approach it at all :( Here is the question:
" Prove or disprove that there is a Linear map $T:\mathbb{R^{3}\rightarrow\mathbb{R^{3}}}$ such that the following occurs:
$T\left(2,1,1\right)=\left(0,3,1\right)$
$T\left(0,3,6\right)=\left(9,2,3\right)$
$T\left(1,2,5\right)=\left(7,3,2\right)$ "
OK I have absolutely no idea how to approach this question. I suppose there is a more elegant way than to sit around and do a trial & error approach to find out the equation for the Linear Map.
Any hints on how to solve it? Thanks!
First, you can check to see whether the vectors $(2,1,1), (0,3,6), (1,2,5)$ are linearly-independent. If they are, then they form a basis for $\mathbb R^3$ . And if they are a basis, knowing the action on them ( meaning their image under $T$ ), defines $T$ fully, and uniquely. And then the matrix representation of $T$ is given by the columns $T(a,b,c)$.