Suppose V and W are finite-dimensional and T∈L(V,W). Prove that there exists a basis of V and a basis of W such that with respect to these bases, all entries of M(T) are 0 except that the entries in row j, column j equal 1 for 1≤j≤dimrange T.
(This generalizes 3c/2)
Let v1,…,vn be a basis of V such that Tv1,…,Tvr is a basis for range T and vr+1,…,vn is a basis for null T. The existance of such a basis is shown in the proof of 3.22 though it isn't in the theorem statement. (todo: ugly! construct it using book results)
Let wj=Tvj for 1≤j≤r then extend to a basis w1,…,wm of W.
Clearly M(T) will have ones in the diagonal up to r=dimrange T since writing Tvk in terms of the matrix, where 1≤k≤r gives
Tvk=j=1∑mAj,kwj
Now since wk=Tvk and the w's are independent we must have
Aj,k={10if j=kotherwise
To finish we need to show that the (r+1),…,m columns are all zero. This is because Tvk=0 for k>r and
Tvk=0=j=1∑mAj,kwj
Implies (by independence of w1,…,wm) that Aj,k=0 for all j.
This was intuitively obvious to me after doing 3c/2, but took a while to make rigorous. I need to improve my "rigorous articulation"