I am currently working on Axler's LADR. I have some difficulties with the following problem 3.D.4:
Problem: Suppose $W$ is finite-dimensional an $T_1, T_2 \in \mathcal{L}(V, W)$. Prove that $null(T_1)=null(T_2)$ if and only if there exists an invertible operator $S\in\mathcal{L}(W)$ sucht that $T_1=ST_2$.
My main problem I guess is, the possible infiniteness of $V$. I came up with the following solution. Note that I present only one direction of the need implications, since the other is pretty straight-forward.
Solution: Suppose that $null(T_1)=null(T_2)$. Since $W$ is finite-dimensional, we know that $range(T_1)$ is finite-dimensional. Let $T_1v_1, ..., T_1v_n$ be a basis of $range(T_1)$. Then there exist vectors $v_1, ..., v_n$ in $V$ that get mapped to $T_1v_1, ..., T_1v_n$. $v_1, ..., v_n$ is linearly independent, because $T_1v_1, ..., T_1v_n$ is linearly independent. Clearly $v_1, ..., v_n\notin null(T_1)$ and we can write any $v\in V$ as $v=a_1v_1 + ... + a_nv_n + u$ where $u\in null(T_1)$.
Since $null(T_1)=null(T_2)$ we have images of $v_1, ..., v_n$ under $T_2$ namely $T_2v_1, ..., T_2v_n$. Now $T_2v_1, ..., T_2v_n$ is spanning $range(T_2)$ and I want to prove, that it also is linearly independent.
Suppose $$T_2c_1v_1 + ... + T_2c_nv_n = 0$$
Then $$T_2(c_1v_1 + ... + c_nv_n) = 0$$
which means that $$c_1v_1 + ... + c_nv_n \in null(T_2)$$
Now if one of the coefficients $c_i$ would be unequal to $0$, then the corresponding $v_i$ would be in $null(T_2)$ and thus also in $null(T_1)$, which we know to be false. Hence $c_1=...=c_n=0$, meaning that $T_2v_1, ..., T_2v_n$ is linearly independent.
So we can extend $T_1v_1, ..., T_1v_n$ and $T_2v_1, ..., T_2v_n$ to bases of $W$ and easily define a $S\in\mathcal{L}(W)$ with the desired properties. $\blacksquare$
Now my questions is, if my line of arguments is correct. I am a little unsure of the statement that $v=a_1v_1 + ... + a_nv_n + u$, which seems a little scetchy due to the possible infiniteness of $V$. And of the deduction, that $T_2v_1, ..., T_2v_n$ is linearly independent. If I am mistaken, is there a possibility of complete the proof? Thanks for any help!