Linear Algebra and Its Applications · Problem 3.1.28

Orthogonality — Gilbert Strang

Mathematics · orthogonal-vectors-subspaces

Problem

This is a system of equations

Ax = b

with no solution:

\begin{aligned} x + 2y + 2z &= 5\\ 2x + 2y + 3z &= 5\\ 3x + 4y + 5z &= 9. \end{aligned}

Find numbers

y_1, y_2, y_3

to multiply the equations so they add to

0 = 1

. You have found a vector

y

in which subspace? The inner product

y^Tb

is 1.