Answer to Question #204534 in Linear Algebra for sabelo Zwelakhe Xu

Question

suppose v are finite dimensional of t € l(v,w). show that with respect to each of bases of v and w, the matrix of t has at least dim range t nonzero entries

Answer 1

Choose bases v₁,v₂,...,v_n of V and w₁,w_{2 ....} .,w_mof W and let M(T) be the matrix of T with respect to these bases. Column j of M(T) will consist entirely of zeros iff Tvj=0; equivalently column j has at least one nonzero entry iff Tv_j≠0. So out of the n basis vectors of V say k≤n of them are mapped to zero; assume without loss that Tv₁,Tv₂,...,Tv_k = 0. Then rangeT=span(Tv_k+1,Tv_k+2,...,Tv_n), which has dimension ≤n−k (since these vectors need not be linearly independent.) Now for each j=k+1,k+2,...,n, Tvj≠0 so at least one of the scalars in its basis expansion with respect to the wi's is nonzero; hence the corresponding column of the matrix has at least one nonzero entry. So each of these n−k vectors contributes at least one nonzero entry to the matrix, so we have at least n−k nonzero entries (which is at least dimrangeT.)