Problem-15

fundamental matrix subspaces

Question

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

Solution

This is not possible as the row space is orthogonal to the nullspace. A proof this statement is given below. By convention all vectors are column vectors, hence row vectors are represented as \(r_i^{T}\).

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

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

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

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

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