# MATH 33AH Lecture Notes - Lecture 13: Diagonalizable Matrix, Linear Map, Symmetric MatrixExam

by OC2716258

Department

MathematicsCourse Code

MATH 33AHProfessor

AllStudy Guide

FinalThis

**preview**shows pages 1-3. to view the full**18 pages of the document.**MATH 33A Final Exam Mar. 19, 2018

STUDENT NAME:

STUDENT ID NUMBER:

DISCUSSION SECTION NUMBER:

Directions

Answer each question in the space provided. Please write clearly and legibly. Show

all of your work—your work must both justify and clearly identify your ﬁnal answer.

No books, notes or calculators are allowed. You must simplify results of function

evaluations when it is possible to do so.

For instructor use only

Page Points Score

2 9

3 12

4 9

5 10

6 6

7 10

8 10

9 10

10 9

11 6

12 9

13 10

14 8

15 10

16 9

17 13

Total: 150

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Final Exam, Page 2 of 18 Mar. 19, 2018

1. [5 pts] Suppose I have a 6x6 matrix Aof real numbers with the following properties:

•Ais not invertible.

•2 + 3iis an eigenvalue for Awith algebraic multiplicity 1.

•4 is an eigenvalue for Awith algebraic multiplicity 2.

•Tr(A) = 11

Write down all of the eigenvalues of Aand give their algebraic multiplicities.

2. [4 pts] Suppose that −→

vand −→

ware two unit vectors with an angle of π

3between them. Find

(−→

v+−→

w)·(2−→

v−−→

w).

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

Final Exam, Page 3 of 18 Mar. 19, 2018

3. [6 pts] Suppose Vis a subspace of Rn, and let PVbe the matrix of orthogonal projection onto

V. Prove that, if PVwas invertible, then in fact V=Rn. In this case, the matrix PVwas a

very special nxnmatrix - what matrix was it?

Hint: Depending on your point of view, it may be easier to prove the following equivalent

statement: If V6=Rn, then PVwas not invertible.

4. (a) [6 pts] Suppose that Vand Ware two subspaces of Rn, and PVand PWare the matrices

of orthogonal projection onto each respective subspace. Prove that, if we assume V⊥W,

then PVPWis the zero matrix.

Hint: Let −→

xbe an arbitrary vector in Rn, and explain why PVPW−→

x=−→

0. Then since −→

x

was arbitrary, conclude that PVPWmust have been the zero matrix.

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