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Response Details:
This proof can get a little lengthy and I am going to assume you have a general understanding of some common elements used in the proof (rank factorization and the like).
Prove: For m x n matrices A and B, Rank(A + B)  Rank(A) + Rank(B).
Proof:
Suppose that  and  . Then there are rank factorization  and  of A and B, where  is  with rank  and  is  with rank  .
 is  with rank and  is  with rank .
Create a  matrix  and a  matrix R by stacking over . Then:
Since the matrix CR is a product, its rank cannot exceed the rank of either of its factors.
Since C has  columns, the rank of C cannot exceed . Likewise, R has rows, so the rank of R cannot exceed . Thus, the rank of A+B cannot exceed  , or the  . QED
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