Find a vector $x\in\Bbb R^2$ s. t. $T(x)=b$.

Question

Let $$A=\begin{bmatrix}1&-3\\3&5\\ -1&7\\\end{bmatrix},b=\begin{bmatrix} 3\\2\\-5\\\end{bmatrix},c=\begin{bmatrix}3\\2\\5\\\end{bmatrix}, u =\begin{bmatrix}2\\-1\\\end{bmatrix}$$ and define a linear transformation of $$T:\Bbb R^2\to\Bbb R^3\ \ \ T(x):= Ax$$

(a) Find $T(u)$

(b) Find $a,b\in\Bbb R^2$ s.t. $T(a)=T(b)=b$.

(c) Is there more than one $x$ s.t. $T(x)=b$?

(d) Determine whether $c\in\operatorname{Im}T$

This question is very frustrating, stressful, and annoying, and I have spent literally $8\mathrm{h}$ trying to solve it, looking up stuff on the internet, as there is NOTHING like it in the note books. All of what we have is $\Bbb R^2\to\Bbb R^2$ with predefined "defined by" values. Asking the professor, he says to "Look it up.", which isn't helpful.

What I know that I have to do is: $T(x+y)=T(x)+T(y)$

and

$T(cx) = cT(x)$

I don't know where to go from there...

Please help.

Thanks.

Answer 1

You know how to multiply a matrix and a vector, so find $Au$ and that is $T(u)$

For the other parts you need to find a vector $v$ such that $Av$ is given.

So you solve for the components of $v$ and see if the answer is unique or even if you have an answer.