Linear Algebra and Its Applications · Problem 3.1.25

Orthogonality — Gilbert Strang

Mathematics · orthogonal-vectors-subspaces

Problem

Construct a matrix with the required property or say why that is impossible. ((a)) Column space contains [12−3]\begin{bmatrix}1\\2\\-3\end{bmatrix} and [2−35]\begin{bmatrix}2\\-3\\5\end{bmatrix}, nullspace contains [111]\begin{bmatrix}1\\1\\1\end{bmatrix}. ((b)) Row space contains [12−3]\begin{bmatrix}1\\2\\-3\end{bmatrix} and [2−35]\begin{bmatrix}2\\-3\\5\end{bmatrix}, nullspace contains [111]\begin{bmatrix}1\\1\\1\end{bmatrix}. ((c)) Ax=[111]Ax = \begin{bmatrix}1\\1\\1\end{bmatrix} has a solution and AT[100]=[000]A^T\begin{bmatrix}1\\0\\0\end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix}. ((d)) Every row is orthogonal to every column (AA is not the zero matrix). ((e)) The columns add up to a column of 0s, the rows add to a row of 1s.

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