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**[SOLVED] Proving col rank = row rank**

**Homework Statement**

Demonstrate these four assertions to get an alternate proof that column rank equals row rank.

(a) [itex]\vec{y}\cdot\vec{y} = \vec{0} \Leftrightarrow \vec{y} = \vec{0}[/itex]

(b) [itex]A\vec{x} = \vec{0} \Leftrightarrow A^TA\vec{x} = \vec{0}[/itex]

(c) [itex]\dim R(A) = \dim R(A^TA)[/itex]

(d) col rank A = col rank [itex]A^T[/itex] = row rank A

**The attempt at a solution**

I don't understand how assertions (a) through (c) are of any importance. The only one that needs demonstrating is (d).

(a) Trivial

(b) Given [itex]A\vec{x} = \vec{0}[/itex], [itex]A^TA\vec{x} = A^T\vec{0} = \vec{0}[/itex]. Given [itex]A^TA\vec{x} = \vec{0}[/itex], suppose [itex]A\vec{x} \ne \vec{0}[/itex]. Then [itex]A^TA\vec{x} \ne A^T\vec{0} = \vec{0}[/itex]. Contradiction.

(c) No clue.

(d) I was thinking of using a "reduce to echelon form" proof, but that just defeats the purpose of this exercise.