Math 1554 Quiz 9 (20mins) Fall 2015

Name: T-square login ID:

This quiz is worth a total of 100 points, and the value of each question is listed with each question.You must show your work for the computational questions; answers without substantiation do not count.

1. Mark each statement True of False. No explanation is required. +5pts each for the correct answerbut -2pts for the incorrect answer .

T / F If ~b is in the orthogonal complement of the column space of A, then a least-squares solution

of A~x = ~b is in the null space of A.

T / F To find y = β0 + β1x of the least-squares line that best fits the given (0,1),(1,3),(2,1),(5,2),

one needs to solve

0 11 32 15 2

[ β0

β1

]=

1111

.T / F There exists a symmetric matrix which is not orthogonally diagonalizable.

T / F If A is symmetric and if vectors ~u and ~v satisfy A~u = ~u and A~v = −~v, then ~u · ~v = 0.

2. (40pts) Construct a spectral decomposition of A =

[2 11 2

].

(continue on next page)

3. Answer the following questions.

A =

1 2 01 2 01 0 31 0 3

, ~b =

1200

.(a) (10pts) Is ~b in the column space of A? Justify your answer.

(b) (20pts) Describe all least-squares solutions of the equation A~x = ~b.

(c) (10pts) Explain why the equation does not have a unique least-squares solution.