13. Suppose $V$ and $W$ are finite-dimensional vector spaces and $T:V \to W$ is an isomorphism. Then there exist bases $\mathcal{B}$ and $\mathcal{C}$, for $V$ and $W$ respectively, such that $[T]_{\mathcal{C},\mathcal{B}}$ is the identity matrix.
14. Let $T:V\to\mathbb{R}$ be a linear transformation. Suppose $\{v_1,\dots,v_n\}$ is a basis for $\ker(T)$. Suppose also that $v \in V$, $v \ne 0$, is not in $\ker(T)$. Prove $\{v,v_1,\dots,v_n\}$ is a basis for $V$.
15. Show that any linear transformation $T:V \to W$ may be written as a sum of linear transformations $T = T_1 + \cdots + T_k$ for some $k$, where each $T_i$ is a linear transformation of rank $1$.
Hey guys, I have a couple of questions I need help with. It'd be great if I could get any sort of help/hints! Thanks.