Let A be an n x n matrix. How do you show that the rows of A are linearly independent if and only if the columns of A span $\displaystyle R^{n}$?
the rows of $\displaystyle A$ are linearly independent iff the columns of $\displaystyle A^T$ are linearly independent iff $\displaystyle A^T$ is invertible iff $\displaystyle A$ is invertible iff the columns of $\displaystyle A$ are linearly independent iff the columns of $\displaystyle A$ span $\displaystyle \mathbb{R}^n.$