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A are linearly dependent, and Ay is a linear combination of the columns of A). Let x be a solution to Ax = b. Then clearly x+y 6= x is also a solution. This contradicts the uniqueness of the solution. Hence, rankA = n. ⇐: By Theorem 2.1, a solution exists. It remains to prove that it is unique. For this, let x and y be solutions, i.e., Ax = b and Ay = b.

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