# Homework Help: Proving row space column space

1. Jun 21, 2011

### nhrock3

A , B are nXn matrices
and
AB=(A)^t
t-is transpose
prove that the space spanned by A's row equals the space spanned by A's columns
i know that there dimentions are equals
so in order to prove equality i need to prove that one is a part of the other
how to do it?

each column i of (AB)_i=A*B_i
i was told by my proff that that column i of AB is a member from the span of the columns of A

but i dont get this result
suppose the member of B_i column is (c1,c2,..,cn)
so the multiplication of A by the B_i column
we get then the first member is dot product from the first row with (c1,c2,..,cn)
i cant see how its a variation from the A columns?

Last edited: Jun 21, 2011
2. Jun 21, 2011

### lanedance

yeah i think you're on the right track,
A.B = A^T

now consider a the kth column B, the vector B_k, which when multiplied with A yields the kth column of A_T, (A^T)_k
A.B_k = (A^T)_k

so the kth column of A^T is a linear combination of the columns of A, given by the components of B_k.