Let A be an m x n matrix and B an n x p matrix. If the columns of A span R m and

Question

Let A be an m x n matrix and B an n x p matrix.
If the columns of A span Rm and the columns of B sapn Rn, do the columns of AB span Rm?

Prove your answer.

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Explanation / Answer

Your new matrix is of the size mxp You will need the concept of rank here, a rank is the number of independent rows or columns a matrix has (It is a thm that they are actually equal) So you are given that rank(A) = m rank(B) = n we want to comment on rank(AB) rank(a matrix) =m, p >= n So we know that A is an map from Rm to Rn (Rank is m so span is a subset of dimension m) also B is a map from Rn to Rp (again span is a subset of dimension n) Hence AB is a composite map from Rm to Rp () Hence atleast Rm must be covered in AB Hence proved. Check my answer of your previous question the guy before me has probably given an incorrect answer

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