The outer product of two column vectors x ∈ Rm, y ∈ Rn, is the m × n matrix x ⊗ y := xy⊤ (the product of a m × 1 matrix with a 1 × n matrix). This is called a rank-one matrix for the following reason: the rank of a matrix A ∈ Rm×n equals 1 if and only if it can be written in the form x y⊤ for nonzero x and y. Prove this fact.

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