Question

Suppose V and W are finite-dimensional and T ∈ L(V, W).

Show that with respect to each choice of bases of V and W, the matrix of T has at least dim range T nonzero entries.

Please assist.

Answer 1

Let $v_1.....v_n$ be a basis of V,$Tv_1,....,Tv_n$ be a basis of range T,$w_1,....w_m$ be a basis of W. Then because each $Tv_j$ is non zero(because the list is lin. Ind.) for each

$Tv_j= A_1,jw_1+........+A_m,jw_m$ ,at least one of $A_i,jw_i$ is non zero ,and so $M(T)$ has at least dim range T non zero entries.