Proving C(AB) = C(A) with Orthogonal Complement and Matrix Multiplication

[samuelr0750]

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 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 $p = b$.

 

[samuelr0750]

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

 

[jbunniii]

By "orthogonal complement of R(A) + C(B)" do you mean $R(A)^{\perp} + C(B)$, or something else?