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Final

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


Department
Mathematics
Course Code
MATH 33AH
Professor
All
Study Guide
Final

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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).

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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 VW,
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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