Let V and W be any two finite dimensional vector spaces over a field F. Let T : V →W be a linear map. Prove that there are bases of V and W such that with respect to those bases T is represented by a matrix composed of 0’s and 1’s only. If T is an isomorphism, prove that with respect to suitable bases it is represented by the identity matrix.
I don't know how to do the first question. For the second question, If $T$ is an isomorphism and there is a basis for which $T$ can be represented as the identity matrix, then $T$ maps all basis vectors to themselves? But it is hard to imagine a rotation in $\mathbb{R}^2$ that fixes any vectors?
Please help
Let $n=\operatorname{rank}T$ and let $f_1,\ldots,f_k\in W$ which span $T(V)$. Add vectors $f_{k+1},\ldots,f_m$ to the set $\{f_1,\ldots,f_k\}$ so that it becames a basis $B^\star$ of $W$. Take $e_1,\ldots,e_k\in V$ such that $T(e_i)=f_i$ end extend $\{e_1,\ldots,e_k\}$ to a basis of $B$ of $V$. Then the matrix $M$ of $T$ with respect to the bases $B$ and $B^\star$ consists only of $0$'s and $1$'s.
Concerning the second problem, it seems that you forgot that you are free to use two distinct bases.