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> Let $A$ be an $m \times n$ matrix. Show that, even though they may be of different sizes, both Gram matrices $K = A^TA$ and $L = AA^T$ have the same rank.
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My attempt:

We have that $K$ and $L$ are Gram matrices so $K = A^TA = (A^TA)^T = AA^T = L$ and by definition we have that $\mathrm{rank}(A) = A^T$.