Linear Algebra and Its Applications · Problem 3.3.27

Orthogonality — Gilbert Strang

Mathematics · statistics-mean-variance

Problem

(Recommended) This problem projects

b = (b_1, \ldots, b_m)

onto the line through

a = (1, \ldots, 1)

. We solve

m

equations

ax = b

in 1 unknown (by least squares). (a) Solve

a^T a \hat{x} = a^T b

to show that

\hat{x}

is the mean (the average) of the

b

's. (b) Find

e = b - a\hat{x}

, the variance

\|e\|^2

, and the standard deviation

\|e\|

. (c) The horizontal line

\hat{b} = 3

is closest to

b = (1, 2, 6)

. Check that

p = (3, 3, 3)

is perpendicular to

e

and find the projection matrix

P