Question-17

Does there exist a matrix whose row space contains \((1, 2, 1)\) and whose nullspace contains \((1, -2, 1)\)?

• Yes
• No

Answer

• Yes
• No

Solution

The row space is orthogonal to the null space. To see why:

• Let \(\displaystyle A=\begin{bmatrix}- & r_{1}^{T} & - \\ & \vdots & \\- & r_{n}^{T}&-\end{bmatrix}\)

• Let \(\displaystyle x\in \mathcal{N}( A) \Longrightarrow Ax=0\)

• Do this component wise. \(\displaystyle r{_{i}}^{T} x=0\) for \(\displaystyle 1\leqslant i\leqslant n\) implying that \(\displaystyle x\) is perpendicular to every row.

• Therefore \(\displaystyle x\) is perpendicular to the row space of \(\displaystyle A\).

• It follows the row space is orthogonal to the null space of \(\displaystyle A\).