Linear Algebra and Its Applications · Problem 3.3.5

Orthogonality — Gilbert Strang

Mathematics · least-squares-inconsistent-system

Problem

The following system has no solution:

Ax = \begin{bmatrix} 1 & -1 \\ 1 & 0 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} C \\ D \end{bmatrix} = \begin{bmatrix} 4 \\ 5 \\ 9 \end{bmatrix} = b.

Sketch and solve a straight-line fit that leads to the minimization of the quadratic

(C - D - 4)^2 + (C - 5)^2 + (C + D - 9)^2

. What is the projection of

b

onto the column space of

A