I was thinking about the following problem:
Let $x,y$ be linearly independent vectors in $\mathbb R^2$. Suppose,$T:\mathbb R^2 \rightarrow \mathbb R^2$ is a linear transformation such that $Ty=\alpha x$ and $Tx=0.$ Then with respect to some basis in $\mathbb R^2$, $T$ is of the form:
(a)\begin{pmatrix}
a & 0\\
0 & a
\end{pmatrix}, where $a>0$
(b)\begin{pmatrix}
a & 0\\
0 & b
\end{pmatrix}, where $a,b>0, a \neq b$
(c)\begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix}
(d)\begin{pmatrix}
0 & 0\\
0 & 0
\end{pmatrix}
I do not know how to proceed with the problem.
If I take $T$ of the form \begin{pmatrix}
a & 0\\
0 & a
\end{pmatrix}, where $a>0$
and then after satisfying the given conditions $Ty=\alpha x$ and $Tx=0,$ i see that $x,y$ are linearly independent.But i want a more direct way of solving it.
Could someone point me in the right direction?Thanks in advance for your time.