UIUC Math 415 M13
Final Exam
May 10, 2013
.
.
Name & UI#:
• This is a closed-book, closed-notes exam. No electronic aids are allowed.
• Read each question carefully. Unless otherwise stated you need to justify your
answer. Do not use results not proven in class.
• Answer the questions in the spaces provided on the question sheets. If you need
extra room, use the back sides of each page. If you must use extra paper, make
sure to write your name on it and attach it to this exam.
• Do not unstaple or detach pages from this exam.
Question Points Score
1
15
2
10
3
7
4
8
5
10
6
10
7
15
8
10
9
15
Total:
100
1. Let A be an n × n matrix.
(a) (4 points) Show that A + AT is a symmetric matrix.
(b) (4 points) Show that A − AT is a skew-symmetric matrix.
(c) (7 points) Prove that A can be written as sum of a symmetric and a skew-symmetric
matrix.
Math 415 M13
Final
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2. (a) (7 points) Let P3 be the linear space of polynomials with degree less than or equal
3 and let D be the differential operator on P3 , that is,
D : P3 −→ P3 given by D(p(x)) = p0 (x).
Find the matrix A of D with respect to standard basis B = (1, x, x2 , x3 ).
(b) (3 points) Consider the basis B = (1, 1+x, 1+x+x2 ) for P2 . Find the B-coordinate
of f (x) = 1 − x − x2 .
Math 415 M13
Final
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3. (7 points) Find a 2 × 2 matrix A such that Ak 6= I2 for k = 1, 2, 3, 4, 5 but A6 = I2 .
4. (8 points) Let A be an n × m matrix. Prove that rank(A)=rank(AT A). State any
theorem you use.
(Hint: In class we showed that ker(A) =ker(AT A).)
Math 415 M13
Final
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
 
1
2



1   1
5. (10 points) Consider the subspace V of R4 given by V = span(
 1  ,  −1
1
0
⊥
a basis for V , the orthogonal complement of V .
Math 415 M13
Final


). Find

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

9 1 0
6. Consider A =  5 0 1 .
0 11 4
(a) (7 points) Find adj(A), the adjoint of A, by using the definition of adjoint.
(b) (3 points) Find the inverse of A.
Math 415 M13
Final
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

0 1 1
7. (15 points) Orthogonally diagonalize the matrix A =  1 0 1  .
1 1 0
(Hint: λ = 2 is an eigenvalue.)
Math 415 M13
Final
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

2 2
8. (a) (7 points) Find the singular values of A =  0 2 
2 0
(b) (3 points) What is the Σ matrix in the singular value decomposition of A above?
Math 415 M13
Final
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9. Select one (you don’t need to justify your answer).
(a) (2 points) Let ~v1 and ~v2 be vectors in Rn such that ~v1 ·~v1 = 9, ~v1 ·~v2 = 10, ~v2 ·~v2 = 25.
Then k~v1 + 2~v2 k equals
√
√
√
√
(b) 149
(c) 117
(d) 159
(e) 59
(a) 151
(b) (2 points) There are vectors ~v1 and ~v2 in Rn such that ~v1 · ~v1 = 4, ~v1 · ~v2 = −7,
~v2 · ~v2 = 9.
(a) True
(b) False


1 2 3 4
(c) (4 points) Consider A =  2 0 6 0 . For which vectors ~b below the
1 2 3 4
~
A~x = b is consistent? (Select two).
 
 
 
 

3
8
1
0
(a)  2 
(b)  2 
(c)  2 
(d)  4 
(e) 
3
7
8
0
(d) (3 points) The matrix A =
(a) [2, ∞)
p 2
2 p
(b) (−∞, 2]
system

8
8 
1
is positive semi-definite when p is in
(c) [−2, 0]
(d) [−2, 2]
√ √
(e) [− 2, 2] .
2×2
(e) (2 points)
Let
including all the matrices that commutes
W be the subspace of R
0 1
with
. Then the dimension of W is
0 1
(a) 0
(b) 1
(c) 2
(d) 3
(e) 4
(f) (2 points) Let A be a 4 × 4 matrix such that fA (λ) = (1 − λ)(2 − λ)3 . Then
(a)
Math 415 M13
det(A) = −7
tr(A) = 8
(b)
det(A) = 8
tr(A) = 7
Final
(c)
det(A) = 8
tr(A) = −7
(d)
det(A) = 7
tr(A) = −8
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