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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.
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.