Exercise 4

Determine whether the following statements are true or false and explain your reasoning.

1. If $A=\begin{bmatrix} 2 & 1 \\ 1 & 1 \\ -3 & 1 \end{bmatrix}$, then $\xvec=\threevec{4}{-5}1$ is in $\col(A)^\perp$.

2. If $A$ is a $2\times3$ matrix and $B$ is a $3\times4$ matrix, then $(AB)^T = A^TB^T$ is a $4\times2$ matrix.

3. If the columns of $A$ are $\vvec_1$, $\vvec_2$, and $\vvec_3$ and $A^T\xvec = \threevec{2}01$, then $\xvec$ is orthogonal to $\vvec_2$.

4. If $A$ is a $4\times 4$ matrix with $\rank(A) = 3$, then $\col(A)^\perp$ is a line in $\real^4$.

5. If $A$ is a $5\times 7$ matrix with $\rank(A) = 5$, then $\rank(A^T) = 7$.