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Linear algebra question

  • #1

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.
 

Answers and Replies

  • #2
jbunniii
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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##.
 
  • #3
I"m sorry, i meant A is nxp and B is p xq.
 
  • #4
jbunniii
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By "orthogonal complement of R(A) + C(B)" do you mean ##R(A)^{\perp} + C(B)##, or something else?
 

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