# Linear algebra question

## Homework Statement

R(M) and C(M) are the row and column spaces of M.
Let A be an nxp matrix, and B be a bxq matrix.
Show that C(AB) = C(A) when the orthogonal complement of R(A) + C(B) = R^p (i.e. the orthogonal complement of R(A) and C(B) span R^p).

## Homework Equations

I know that the orthogonal complement of R(A) is the null spaceo f A.
I also know that C(X'X) = C(X') but that doesn't help

## The Attempt at a Solution

Not sure where to go... thanks.

jbunniii
Homework Helper
Gold Member

## Homework Statement

R(M) and C(M) are the row and column spaces of M.
Let A be an nxp matrix, and B be a bxq matrix.
Show that C(AB) = C(A) when the orthogonal complement of R(A) + C(B) = R^p (i.e. the orthogonal complement of R(A) and C(B) span R^p).
Are you sure you stated the question correctly? If ##A## is ##n \times p## and ##B## is ##b \times q##, then the product ##AB## isn't even defined unless ##p = b##.

I"m sorry, i meant A is nxp and B is p xq.

jbunniii