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1 Suppose V and W are both finite-dimensional. Prove that there exists an injective linear map from V to W if and only if dim V ≤ dim W. (Hint for 'only if' part: use rank-nullity theorem.) (a) Prove that if A is a 3 × 1 matrix and B is a 1 × 3 matrix, then the 3 × 3 matrix AB has rank at most 1. (b) Conversely, show that if C is any 3 × 3 matrix having rank 1, then there exist a 3 × 1matrix A and a 1 × 3 matrix B such that C = AB. (Hint: What can you say for the rows of a 3 × 3 matrix with rank 1?)

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