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Linear Algebra Final Exam Practice Problems and Answers, Exams of Algebra

Practice problems and answers for a Linear Algebra final exam. The problems cover topics such as matrices, determinants, bases, kernel, and range. The answers are provided for each problem. from a summer 2015 session at an unknown university.

Typology: Exams

2014/2015

Uploaded on 05/11/2023

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Final Exam Practice Problems โ€” Answers
Math 240 โ€” Calculus III
Summer 2015, Session II
Linear Algebra
1. (a) {(0,0) โˆˆR2}
(b) {(โˆ’t, 4t, t)โˆˆR3:tโˆˆR}
(c) {(1,2,4) โˆˆR3}
(d) This system is inconsistent.
(e) {(โˆ’1โˆ’2sโˆ’4t, s, โˆ’2โˆ’3t, t)โˆˆR4:s, t โˆˆR}(Answers may vary.)
2. x(t) = (3 โˆ’2t, 1โˆ’t, t)
3. (a) โˆ’7
(b) 29
(c) โˆ’2
4. (a) The inverse of this matrix is
1
2๎˜”2โˆ’4
โˆ’1 3๎˜•.
(b) This matrix is not invertible.
(c) The inverse of this matrix is
๏ฃฎ
๏ฃฐ
โˆ’3 0 โˆ’4
โˆ’2 1 โˆ’3
1 0 1
๏ฃน
๏ฃป.
5. (a) Answers may vary. Correct answers include
{v1,v3}and {(1,2,1,3),(0,1,4,1)}.
(b) Answers may vary. Correct answers include
{v1,v2,v4}and {(1,1,1,1,1),(0,0,1,3,0),(0,0,0,1,0)}.
6. One method of verification is to compute
det ๎˜€๎˜‚v1v2๎˜ƒ๎˜=๎˜Œ
๎˜Œ
๎˜Œ
๎˜Œ
1 1
1 2๎˜Œ
๎˜Œ
๎˜Œ
๎˜Œ
= 1
and note that it is nonzero. Then
(2,โˆ’1) = 5v1โˆ’3v2.
pf3
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Final Exam Practice Problems โ€” Answers

Math 240 โ€” Calculus III

Summer 2015, Session II

Linear Algebra

  1. (a) {(0, 0) โˆˆ R^2 }

(b) {(โˆ’t, 4 t, t) โˆˆ R^3 : t โˆˆ R}

(c) {(1, 2 , 4) โˆˆ R^3 }

(d) This system is inconsistent.

(e) {(โˆ’ 1 โˆ’ 2 s โˆ’ 4 t, s, โˆ’ 2 โˆ’ 3 t, t) โˆˆ R^4 : s, t โˆˆ R} (Answers may vary.)

  1. x(t) = (3 โˆ’ 2 t, 1 โˆ’ t, t)
  2. (a) โˆ’ 7

(b) 29

(c) โˆ’ 2

  1. (a) The inverse of this matrix is 1

2

[

]

(b) This matrix is not invertible.

(c) The inverse of this matrix is (^) ๏ฃฎ

  1. (a) Answers may vary. Correct answers include

{v 1 , v 3 } and {(1, 2 , 1 , 3), (0, 1 , 4 , 1)}.

(b) Answers may vary. Correct answers include

{v 1 , v 2 , v 4 } and {(1, 1 , 1 , 1 , 1), (0, 0 , 1 , 3 , 0), (0, 0 , 0 , 1 , 0)}.

  1. One method of verification is to compute

det

([

v 1 v 2

])

and note that it is nonzero. Then

(2, โˆ’1) = 5v 1 โˆ’ 3 v 2.

  1. A basis for Ker(T ) is {(โˆ’ 2 , 1 , 1)}. A basis for Rng(T ) is {(1, โˆ’ 2 , 3), (0, 1 , โˆ’2)}. (Answers may

vary.)

  1. (a)

[

]

(b)

[

]

[

]

  1. (a) False.

(b) True.

(c) True.

(d) True.

(e) False.

More Linear Algebra

11. [D] =

๏ฃป ,^ [L] =

  1. x โˆ’ 2 y + z = 0
  2. (a) 3

[

]

[

]

[

]

[

]

(b) (x^3 + 7x + 3) โˆ’ (x^3 + 7x^2 + 3) โˆ’ (x^3 โˆ’ x^2 โˆ’ x) + (x^3 + 6x^2 โˆ’ 8 x) = 0

  1. (a) This is a basis.

(b) Not a basis.

(c) Not a basis โ€” C^0 (R) is infinite dimensional.

  1. We can see from the definition that T takes a 2 ร— 2 matrix as input and produces a 2 ร— 3

matrix. Now we check that it preserves addition and scalar multiplication:

T

([

a 1 b 1

c 1 d 1

]

[

a 2 b 2

c 2 d 2

])

= T

([

a 1 + a 2 b 1 + b 2

c 1 + c 2 d 1 + d 2

])

[

a 1 + a 2 a 1 + a 2 a 1 + a 2 โˆ’ b 1 โˆ’ b 2

c 1 + c 2 c 1 + c 2 c 1 + c 2 โˆ’ d 1 โˆ’ d 2

]

  1. x(t) = (^12)

[

e^3 t^ + eโˆ’t

5 e^3 t^ + eโˆ’t

]

  1. x(t) = c 1 et

[

cos 2t

cos 2t + sin 2t

]

  • c 2 et

[

sin 2t

sin 2t โˆ’ cos 2t

]

  1. x(t) =

1 5 e

โˆ’t

[

5 cos t 2 cos t + sin t

]

3 5 e

โˆ’t

[

5 sin t 2 sin t โˆ’ cos t

]

= eโˆ’t

[

cos t โˆ’ 3 sin t cos t โˆ’ sin t

]

  1. x(t) = c 1 et

[

]

  • c 2 et

[

2 t + 1

t

]

= et

[

2 c 1 + c 2 (2t + 1)

c 1 + c 2 t

]

  1. x(t) = c 1

[

]

  • c 2

[

3 t + 1

โˆ’t

]

[

3 c 1 + c 2 (3t + 1)

โˆ’c 1 โˆ’ c 2 t

]

  1. x(t) = c 1 eโˆ’^2 t

๏ฃป (^) + c 2 eโˆ’t

2 cos

2 t

sin

2 t โˆš 2 cos

2 t โˆ’ sin

2 t

๏ฃป (^) + c 3 eโˆ’t

2 sin

2 t

โˆ’ cos

2 t โˆš 2 sin

2 t + cos

2 t

  1. x(t) = c 1 eโˆ’^2 t

๏ฃป (^) + c 2 eโˆ’^2 t

๏ฃป (^) + c 3 eโˆ’^2 t

4 t + 1

โˆ’ 2 t โˆ’ 4 t

  1. (a) y(x) = c 1 x^2 + c 2 xโˆ’^4

(b) y(x) = c 1 + c 2 x

โˆš (^7) + c 3 x

โˆ’

โˆš 7

  1. (a) A(D) = D^2 (D โˆ’ 1)

(b) A(D) = (D โˆ’ 7)^4 (D^2 + 16)

(c) A(D) = (D โˆ’ 4)^2 (D^2 โˆ’ 8 D + 41)D^2 (D^2 + 4D + 5)^3

(d) A(D) = D^2 + 6D + 10

  1. (a) y(x) = c 1 eโˆ’^2 x^ + c 2 xeโˆ’^2 x

(b) y(x) = c 1 eโˆ’^4 x^ cos 2x + c 2 eโˆ’^4 x^ sin 2x

(c) y(x) = c 1 eโˆ’^2 x^ + c 2 e^2 x^ + c 3 xe^2 x

(d) y(x) = c 1 eโˆ’x^ + c 2 e^2 x^ + 53 e^2 x

(e) y(t) = c 1 cos 2t + c 2 sin 2t +

32 t^ โˆ’^

1 12 t

3 )^ cos 2t + (^7 4 t^ +^

13 16 t

2 )^ sin 2t

(f) y(t) = c 1 eโˆ’^2 t^ + c 2 teโˆ’^2 t^ + 16 t^3 eโˆ’^2 t^ โˆ’ 18 e^2 t

  1. (a) y(t) = (2 โˆ’ t)e^4 t

(b) y(t) = et^ โˆ’ cos t

  1. (a) y(t) = eโˆ’t^ (cos 2t + 2 sin 2t) =

5 eโˆ’t^ cos(2t โˆ’ arctan 2). This system is underdamped.

(b) y(t) = โˆ’ 14 eโˆ’t/^2 + 54 eโˆ’^5 t/^2 = eโˆ’^3 t/^2

โˆ’ 14 et^ + 54 eโˆ’t

. This system is overdamped.

(c) y(t) = eโˆ’^2 t^ โˆ’ 2 eโˆ’^3 t^ = eโˆ’^5 t/^2

et/^2 โˆ’ 2 eโˆ’t/^2

. This system is overdamped.

  1. y(t) = 16 cos 34 t + 12 sin 34 t โˆ’ 16 cos 2t
  2. (a) y(x) = (c 1 + c 2 t)e^3 t^ + 4t^5 /^2 e^3 t

(b) y(x) = c 1 ex^ + c 2 xex^ โˆ’ ex^ ln x