Exam/Review for Ch 5 and 6

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

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

1) A = –24 –14

84 46

, Ώ = 4 1)

A)

1

–2

B)

10

C)

–2

1

D)

1

46

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

2) A =

1 – 4 – 4

– 4 1 4

4 – 4 – 7

, Ώ = –3 2)

A)

101

,

01

–1

B)

10

–1

C)

01

–1

D)

10

–1

,

011

Find the characteristic equation of the given matrix.

3) A =

1 –7 4 9

0 –5 7 –1

0 0 –7 5

0 0 0 6

3)

A) (6 – Ώ)(5 – Ώ)(–1 – Ώ)(9 – Ώ) = 0 B) (1 – Ώ)(–5 – Ώ)(–7 – Ώ)(6 – Ώ) = 0

C) (9 – Ώ)(–1 – Ώ)(5 – Ώ)(6 – Ώ) = 0 D) (1 – Ώ)(–7 – Ώ)(4 – Ώ)(9 – Ώ) = 0

Find the eigenvalues of the given matrix.

4) 0 –1

2 3

4)

A) –2 B) 1 C) 1, 2 D) 1, –2

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

5) Ώ5 + 17Ώ4 + 72Ώ3 5)

A) 0 (multiplicity 3), 8 (multiplicity 1), 9 (multiplicity 1)

B) –9 (multiplicity 1), –8 (multiplicity 1)

C) 0 (multiplicity 1), – –

D) 8 (multiplicity 1), 9 (multiplicity 1)

1

Find a formula for Ak, given that A = PDP–1, where P and D are given below.

6) A = 5 3

–2 10

, P = 3 1

2 1

, D = 7 0

0 8

6)

A)

3 · 7k – 2 · 8k 3 · 8k – 3 · 7k

2 · 7k – 2 · 8k 3 · 8k – 2 · 7k

B)

3 · 7k + 2 · 8k 3 · 8k + 3 · 7k

2 · 7k + 2 · 8k 3 · 8k + 2 · 7k

C)

3 · 7k – 2 · 8k 3 · 8k + 3 · 7k

2 · 7k + 2 · 8k 3 · 8k – 2 · 7k

D)

7k 0

0 8k

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

7) A =

1 1 4

0 –4 0

–5 –1 –8

7)

A)

P =

1 0 –1

0 –4 0

1 1 1

, D =

–4 0 0

0 1 0

0 0 –3

B)

P =

1 0 –1

–9 –4 0

1 1 1

, D =

–4 0 –3

0 –4 0

0 –4 –3

C)

P =

1 –9 –1

–9 –4 0

1 –4 1

, D =

–4 1 0

0 –4 0

0 0 –3

D)

P =

1 0 –1

–9 –4 0

1 1 1

, D =

–4 0 0

0 –4 0

0 0 –3

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

8) 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) = –5c1 – 6c2 + 5c3

T(b2) = –5c1 – 12c2 + 2c3

8)

A)

–5 0

–6 –6

5 –3

B)

–5 –6 5

0 6 3

C)

–5 –6 5

–5 –12 2

D)

–5 –5

–6 –12

5 2

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.

9) Find a basis B for R2 and the B–matrix D for T with the property that D is a diagonal matrix.

A = –67 –60

72 65

9)

A)

B = 1

–1

,

5

–6

, D = –7 0

0 5

B)

B = 1

5

,

–1

6

, D = –7 0

0 5

C)

B = 1

–1

,

56

, D = –7 0

0 5

D)

B = 5

–6

,

1

–1

, D = –7 0

0 5

2

Compute the dot product u · v.

10) u =

–1

3

3

, v =

5

2

–3

10)

A) 8 B) –8 C) 0 D) –2

Find a unit vector in the direction of the given vector.

11) 16

–32

11)

A)

1

3

–

2

3

B)

15

–

25

C)

1

5

–

2

5

D)

1

5

2

5

Determine whether the set of vectors is orthogonal.

12)

363

,

–3

0

3

,

3

–3

3

12)

A) Yes B) No

13)

20

40

20

,

–20

0

20

,

20

20

20

13)

A) Yes B) No

Find the distance between the two vectors.

14) u = (0, 0, 0) , v = (6, 9, 9) 14)

A) 24 B) 2 6 C) 3 22 D) 198

Express the vector x as a linear combination of the u’s.

15) u1 =

–2

0

1

, u2 =

356

, u3 =

–2

6

–4

, x =

4

–14

33

15)

A) x = –5u1 + 4u2 + 4u3 B) x = 5u1 + 2u2 – 4u3

C) x = –5u1 – 2u2 + 4u3 D) x = 10u1 + 4u2 – 8u3

Find the orthogonal projection of y onto u.

16) y = –3

–4

, u = 5

–10

16)

A)

1

–2

B)

5

–10

C)

25

–50

D)

15

–

25

3

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

17) y =

12

14

25

, u1 =

2

2

–1

, u2 =

–1

3

4

17)

A)

y =

1

21

17

+

11

–7

8

B)

y =

2

42

34

+

10

–28

–9

C)

y =

1

21

17

+

–11

7

–8

D)

y =

1

21

17

+

13

35

42

Find the new coordinate vector for the vector x after performing the specified change of basis.

18) Consider two bases B = b1, b2 and C = c1, c2 for a vector space V such that

b1 = c1 – 6c2 and b2 = 4c1 + 3c2. Suppose x = b1 + 2b2. That is, suppose [x]B = 1

2

. Find [x]C.

18)

A)

6

–9

B)

90

C)

80

D)

–11

10

Find the specified change–of–coordinates matrix.

19) Consider two bases B = b1, b2 and C = c1, c2 for a vector space V such that

b1 = c1 – 2c2 and b2 = 3c1 – 4c2. Find the change–of–coordinates matrix from B to C.

19)

A)

1 3

–2 –4

B)

1 3

2 4

C)

1 –2

3 –4

D)

0 3

–2 –4

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

20) y =

10

20

33

, u1 =

2

2

–1

, u2 =

–1

3

4

20)

A)

–1

27

25

B)

11

23

5

C)

–8

96

95

D)

1

–27

–25

4

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