Exercise 10

Given an $m\times n$ matrix $A$, the row space of $A$ is the column space of $A^T$; that is, $\row(A) = \col(A^T)$.

1. Suppose that $A$ is a $7\times 15$ matrix. For what $p$ is $\row(A)$ a subspace of $\real^p$?

2. How can this proposition help us describe $\row(A)^\perp$?

3. Suppose that $A = \begin{bmatrix} -1 & -2 & 2 & 1 \\ 2 & 4 & -1 & 5 \\ 1 & 2 & 0 & 3 \end{bmatrix}$. Find bases for $\row(A)$ and $\row(A)^\perp$.