This is question 3 of Exercise 3.C in Axler's Linear Algebra Done Right:
Suppose 𝑉 and 𝑊 are finite-dimensional and 𝑇 is a linear map from 𝑉 to 𝑊. Prove that there exist a basis of 𝑉 and a basis of 𝑊 such that with respect to these bases, all entries of the matrix of the linear map 𝑇, 𝑀(𝑇), are 0 except that the entries in row 𝑗, column 𝑗, equal 1 for 1≤𝑗≤dim(𝑟𝑎𝑛𝑔𝑒𝑇)
I have seen that there are a couple of answers on this site (for example this one), but I am still struggling to get a clear and understandable solution that is also in the notation and follows the order of Axler's text. I understand that a matrix is always with respect to bases from the vector spaces involved…is the question asking me to give examples of the two bases? I'm just very perplexed.
Best Answer
Let's prepare our setting:
1- Let $dimV=n$ and $dimW=m$.
2- Let $dim rangeT= k$ and $dim nullT = s$. Note that, by "Rank-Nullity Theorem"(3.22 in Axler's Book), $k+s=n$.
Step1 : Let $B_{1}=(a_{1},a_{2},...,a_{s})$ be a basis of nullT.
Step2: Extend $B_{1}$ to a basis of $V$, say, $B=(v_{1},v_{2},..v_{k},a_{1},a_{2},...,a_{s})$.
Step3: Using the idea in proof of (3.22), we know that $B_{2}=(Tv_{1},Tv_{2},...,Tv_{k})$ is a basis of rangeT.
Step4: Similar to step2, we can extend $B_{2}$ to a basis of $W$, which is $B^{'}=(Tv_{1},Tv_{2},...,Tv_{k},w_{k+1},...,w_{m})$.
Step5: Now, you need to observe that $M(T,B,B^{'})$ is in the desired form.