Exam Name - University of California, San Diegobdriver/math20e_F2010/Notes/final_practice.pdfExam...

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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

For the given matrix A, find a basis for the corresponding eigenspace for the given eigenvalue.

1) A = 4 0 0

27 -5 0

102 -28 2

, λ = 4

A)

1

3

0

, 1

0

9

B)

1

3

0

, 1

0

-9

C)

1

3

9

D)

1

-3

-9

1)

For the given matrix and eigenvalue, find an eigenvector corresponding to the eigenvalue.

2) A = -10 -1

56 5, λ = -2

A)

1

5

B)

-8

1

C)

1

0

D)

1

-8

2)

Find the eigenvalues of the given matrix.

3) -7 -4

33 16

A) -4 B) 4, 5 C) 5 D) -4, -5

3)

The characteristic polynomial of a 5 × 5 matrix is given below. Find the eigenvalues and their multiplicities.

4) λ5 - 21λ4 + 135λ3 - 243λ2

A) 0 (multiplicity 2), -9 (multiplicity 2), 3 (multiplicity 1)

B) 0 (multiplicity 2), 9 (multiplicity 2), 3 (multiplicity 1)

C) 0 (multiplicity 1), 9 (multiplicity 3), 3 (multiplicity 1)

D) 0 (multiplicity 2), -9 (multiplicity 2), -3 (multiplicity 1)

4)

Diagonalize the matrix A, if possible. That is, find an invertible matrix P and a diagonal matrix D such that A= PDP-1.

5) A =

-6 0 0 0

0 -6 0 0

1 -4 6 0

-1 2 0 6

A)

P =

12 24 0 0

6 6 0 0

1 0 1 0

0 1 0 1

, D =

-6 0 0 0

0 -6 0 0

0 0 6 0

0 0 0 6

B) Not diagonalizable

C)

P =

12 24 0 0

-6 -6 0 0

1 0 1 0

0 1 0 1

, D =

6 0 0 0

0 6 0 0

0 0 -6 0

0 0 0 -6

D)

P =

12 -6 1 0

24 -6 0 0

0 0 1 0

0 0 0 1

, D =

6 0 0 0

0 6 0 0

0 0 -6 0

0 0 0 -6

5)

1

6) A = -11 3 -9

0 -5 0

6 -3 4

A)

P = 1 5 -1

5 3 0

1 3 1

, D = -5 1 0

0 -5 0

0 0 -2

B)

P = 1 0 -1

5 3 0

1 1 1

, D = -5 0 -2

0 -5 0

0 -5 -2

C)

P = 1 0 -1

0 3 0

1 1 1

, D = -5 0 0

0 1 0

0 0 -2

D)

P = 1 0 -1

5 3 0

1 1 1

, D = -5 0 0

0 -5 0

0 0 -2

6)

Find the matrix of the linear transformation T: V →W relative to B and C.

7) Suppose B = {b1, b2} is a basis for V and C = {c1, c2, c3} is a basis for W. Let T be defined by

T(b1) = 2c1 + 3c2 - 2c3

T(b2) = 2c1 + 6c2 + 8c3

A)

2 3 -2

2 6 8

B)

2 2

3 6

-2 8

C)

2 3 -2

0 -3 -10

D)

2 0

3 3

-2 10

7)

Define T: R2 → R2 by T(x) = Ax, where A is the matrix defined below. Find the requested basis B for R2 and the

corresponding B-matrix for T.

8) Find a basis B for R2 and the B-matrix D for T with the property that D is an upper triangular

matrix.

A = 23 -81

4 -13

A)

B = -9

2, -4

1, D = -5 1

0 -5

B)

B = -9

4, -2

1, D = 5 1

0 5

C)

B = -9

-2, 4

1, D = 5 1

0 5

D)

B = -9

-2, 4

1, D = 5 1

0 6

8)

Compute the dot product u · v.

9) u = 5

0, v = 15

-10

A) -50 B) 85 C) 65 D) 75

9)

10) u = -14

8, v = 0

11

A) -154 B) 74 C) 88 D) 102

10)

2

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Find the orthogonal projection of y onto u.

11) y = -24

-10, u = 3

-15

A)

1/3

-5/3

B)

3

-15

C)

9

-45

D)

1

-5

11)

Determine whether the set of vectors is orthogonal.

12)12

24

12

, -12

0

12

, 12

-12

12

A) Yes B) No

12)

Let W be the subspace spanned by the uʹs. Write y as the sum of a vector in W and a vector orthogonal to W.

13) y = 11

9

22

, u1 = 2

2

-1

, u2 = -1

3

4

A)

y = 0

16

14

+ 11

-7

8

B)

y = 0

16

14

+ 11

25

36

C)

y = 0

32

28

+ 11

-23

-6

D)

y = 0

16

14

+ -11

7

-8

13)

Find the closest point to y in the subspace W spanned by u1 and u2.

14) y = 12

-1

2

, u1 = 1

0

-1

, u2 = 2

1

2

A)

11

3

1

B)

-11

-3

-1

C)

13

5

7

D)

20

9

16

14)

Find a QR factorization of the matrix A.

15) A = -6 6

3 -18

0 6

A)

Q = -6 -6

3 -12

0 6

, R =

-15

5

30

5

036

6

B)

Q =

2

5-

1

6

- 1

5-

2

6

01

6

, R =

-15

5

30

5

036

6

C)

Q =

2

5-

1

6

- 1

5-

2

6

01

6

, R = -3 6

0 6

D)

Q = -6 6

3 -18

0 6

, R =

30

5 0

36

6

-15

5

15)

3

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The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.

16) Let x1 =

0

1

-1

1

, x2 =

1

1

-1

-1

, x3 =

1

0

1

1

A)

0

1

-1

1

,

1

0

0

-2

,

6

0

1

3

B)

0

1

-1

1

,

3

2

-2

-4

,

14

2

9

7

C)

0

1

-1

1

,

3

4

-4

-2

,

18

4

19

13

D)

0

1

-1

1

,

1

1

-1

-1

,

1

0

1

1

16)

Find a least-squares solution of the inconsistent system Ax = b.

17) A = 1 2

3 4

5 9

, b = 1

5

2

A)

133

3508

- 67

3508

B)

- 17

42

- 67

42

C)

133

27

- 67

27

D)

18

7

- 11

7

17)

Given A and b, determine the least-squares error in the least-squares solution of Ax = b.

18) A = 4 3

2 1

3 2

, b = 3

1

1

A) 66.7591026 B) 186.822034 C) 0.81649658 D) 2.84800125

18)

Determine whether the matrix is symmetric.

19) 1 7

4 0

A) Yes B) No

19)

Orthogonally diagonalize the matrix, giving an orthogonal matrix P and a diagonal matrix D.

20)13 10 4

10 12 6

4 6 5

A)

P = 2 -2 1

2 1 -2

1 2 2

, D = 25 0 0

0 4 0

0 0 1

B)

P = 2/3 -2/3 1/3

2/3 1/3 -2/3

1/3 2/3 2/3

, D = 1 0 0

0 4 0

0 0 25

C)

P = 2/3 -2/3 1/3

2/3 1/3 -2/3

1/3 2/3 2/3

, D = 25 0 0

0 4 0

0 0 1

D)

P = 2 -2 1

2 1 -2

1 2 2

, D = 1 0 0

0 4 0

0 0 25

20)

4

Answer KeyTestname: FINAL_PRACTICE

1) CObjective: (5.1) Find Basis of Eigenspace Given Eigenvalue

2) DObjective: (5.1) Find Eigenvector Given Eigenvalue

3) BObjective: (5.2) Find Eigenvalues of 2 x 2 Matrix

4) BObjective: (5.2) Find Eigenvalues and Multiplicities Given Char Poly

5) AObjective: (5.3) Diagonalize Matrix, if Possible (4 x 4)

6) DObjective: (5.3) Diagonalize Matrix, if Possible (3 x 3)

7) BObjective: (5.4) Find Matrix of Linear Transformation

8) CObjective: (5.4) Find Basis and Matrix of Linear Transformation

9) DObjective: (6.1) Compute Dot Product of Two Vectors

10) CObjective: (6.1) Compute Dot Product of Two Vectors

11) DObjective: (6.2) Find Orthogonal Projection

12) AObjective: (6.2) Determine if Set of Vectors is Orthogonal (Y/N)

13) AObjective: (6.3) Write Vector As Sum of Two Vectors

14) AObjective: (6.3) Find Point in Subspace Closest to Vector

15) BObjective: (6.4) Find QR Factorization of Matrix

16) BObjective: (6.4) Construct Orthogonal Basis for Subspace

17) CObjective: (6.5) Find Least-Squares Solution

18) CObjective: (6.5) Determine Least-Squares Error

19) BObjective: (7.1) Determine if Matrix is Symmetric (Y/N)

20) CObjective: (7.1) Orthogonally Diagonalize 3 x 3 Matrix

5

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