I can't put the whole question on the title part due to the limitation of characters. It actually looks like-
Let, $V$ and $W$ be two vector spaces such that $\text{dim}V=\text{dim}W$ and $T:V\to W$ be a linear map.
Then show that $\exists$ ordered bases $\beta$ and $\gamma$ for $V$
and $W$, respectively, such that $[T]^{\gamma}_{\beta}$ is a diagonal
matrix.
I have tried it, but I can't reach any proper solution.
I first choose any arbitrary basis $\beta=\{v_1, v_2, …,v_n\}$ for $V$. Then we know the set $\{T(v_1), T(v_2), …, T(v_n)\}$ spans $T(V)$. Then I have extracted a linear independent set (say $\gamma^{\prime})$ from $\{T(v_1), T(v_2), …, T(v_n)\}$. Now extend it to a basis of $W$, I call it $\gamma$.
BUT in this way I can't say anything about $[T]^{\gamma}_{\beta}$
Can anybody help me with a proper way out to the problem? Thanks for assistance in advance.
Best Answer
Let $\{v_1,\dots,v_n\}$ be a basis of $\ker T$ in $V$. Extend it to an ordered basis $\beta=\{v_1,\dots,v_n,w_1,\dots,w_m\}$ of $V$. Then $\{T(w_1),\dots,T(w_m)\}$ is a basis of $\operatorname{Im}(T)$ in $W$. Extend it to an ordered basis $\gamma=\{u_1,\dots,u_n,T(w_1),\dots,T(w_m)\}$ of $W$. Then $[T]_\beta^\gamma$ is diagonal, and in fact simply has lower right block $I_m$, and $0$ elsewhere.