Exercise 27

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

• $\vec{y}\cdot\vec{y}=0$ iff $\vec{y}=\vec{0}$.

• $A\vec{x}=\vec{0}$ iff ${{A}^{\mathsf{T}}}A\vec{x}=\vec{0}$.

• $\dim(\mathscr{R}(A))=\dim(\mathscr{R}({{A}^{\mathsf{T}}}A))$.

• $\text{col rank}(A)=\text{col rank}({{A}^{\mathsf{T}}})=\text{row rank}(A)$.