$\newcommand{\ones}{\mathbf 1}$

If $A^T B = 0$, then

Consider the set of vectors $S = \left\lbrace \left[ \begin{array}{r} 1 \\ 0 \\ -1 \end{array} \right], \; \left[ \begin{array}{c} 0 \\ 1 \\ 2 \end{array} \right] \right\rbrace$.

$S$ is a basis for $\mathbf{R}^3$.

$S$ is independent.

The angle between vectors $a = \left[ \begin{array}{c} 1 \\ 2 \\ 3 \end{array} \right]$ and $b=\left[ \begin{array}{r} -1 \\ -2 \\ 3 \end{array}\right]$ is:

$\left[ \begin{array}{c} 1 \\ 3 \\ 4 \end{array} \right]$ is in the range of $\left[\begin{array}{cc} 1 & 0 \\ 2 & 1 \\ 3 & 1 \end{array}\right]$.

• Correct! Result of matrix multiplication from the right by the vector $\left[ \begin{array}{c} 1 \\ 1 \end{array} \right]$.
• Incorrect.

The matrix $\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right]$ has a left inverse.