Suppose and are finite-dimensional and . Show that with respect to each choice of bases of and , the matrix of has at least nonzero entries.
Let and be the bases for and that will be used for the matrix (we don't control these).
Let be the matrix of , we can clearly find at least vectors in with . This implies there is a nonzero element in each row because
Implies at least one (independence of basis). doing times implies the number of nonzero entries is at least .
This is the best bound we can get using only the dimension of the range, since the identity matrix has the number of nonzero entries equal to the dimension of the range.
todo (easy): Remove "clearly" and make rigorous, but make sure it's still understandable!