Linear Algebra and Its Applications · Problem 3.1.35

Orthogonality — Gilbert Strang

Mathematics · orthogonal-subspaces

Problem

The floor and the wall are not orthogonal subspaces because they share a nonzero vector (along the line where they meet). Two planes in

\mathbf{R}^3

cannot be orthogonal! Find a vector in both column spaces

C(A)

and

C(B)

A = \begin{bmatrix}1 & 2\\1 & 3\\1 & 2\end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix}5 & 4\\6 & 3\\5 & 1\end{bmatrix}.

This will be a vector

Ax

and also

B\hat{x}

. Think 3 by 4 with the matrix

[A\ B]