I have the following question:
But I do not know how to start solving it, especially because as we know that representation of a linear transformation is not that easy. shall I use the matrix representation of a linear transformation?
I am unable to solve this problem, could anyone help me in doing so, please? giving example of how to proceed will be very helpful.
Hint for the reverse inclusion of $\boldsymbol{(2)}$:
One does not have in general $f(A)\cap f(B)\subseteq f(A\cap B)$.
Consider the $x y$-plane, and two distinct lines $\ell,\ell'\,$ through the origin, an take for the linear map $\alpha$ the projection onto the $x$-axis. What are $\alpha(\ell)$ and $\alpha(\ell')$? What is $\ell\cap\ell'$?