The simplest way to determine the matrix entries of a linear transfromation, in a given basis, is to apply the linear transformation to each basis vector in turn. Each gives one column of the matrix. For example, this maps each of the first m-1 basis vectors into itself so n-1 columns just give the identity matrix with an addtional "0" on the bottom. , however, is mapped into : The column vector with all "0"s except that the last entry is "1" is mapped into a column vector with all "0"s except that the lasttwoentries are "1". The final column of the matrix is all "0"s except that the last two entries are "1".

Okay, what(b)Suppose here that and such that for all . Prove that is linearly dependent.

A hint was provided as follows: What is the dimension of ?

I'll try to do some of this, but there's a lot of info that I'm unfamiliar with (primarily the change of basis parts).isthe dimension of W? It cannot be any larger than the dimension of F[x] can it? And a basis is thelargestpossible independent set of vectors.