Linear Algebra and Its Applications · Problem 3.3.3

Orthogonality — Gilbert Strang

Mathematics · least-squares-projection

Problem

Solve

Ax = b

by least squares, and find

p = A\hat{x}

A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}.

Verify that the error

b - p

is perpendicular to the columns of

A