MATH 3330

INFORMATION SHEET FOR FINAL EXAM

FALL 2011

FINAL EXAM will be in PKH 103 at 2:00-4:30 pm on Tues Dec 13
• See above for date, time and location of FINAL EXAM. Recall from the first-day handout that any student not obtaining a positive score on the FINAL EXAM will not pass this class.
• The material covered will be the same as that covered on the homework from the start of the semester through Dec 6 (but not §6.3) inclusive. (Homework is listed at my website: www.uta.edu/math/vancliff/T/F11 .)
• My remaining office hours are: 3:30-4:20 pm on Thurs Dec 8 and 3:30-5:30 pm on Mon Dec 12.
• This test will be, in part, multiple choice, but you do NOT need to bring a scantron form. There will be several choices of answer per multiple-choice question and, for each, only one answer will be the correct one. You should do rough work on the test or on paper provided by me. No calculator is allowed. No notes or cards are allowed.
BRING YOUR MYMAV ID CARD WITH YOU.
• When I write a test, I look over the lecture notes and homework which have already been assigned, and use them to model about 85% of the test problems (and most of them are fair game). You should expect between 30 and 40 questions in total.
• A good way to review is to go over the homework problems you have not already done & make sure you understand all the homework well by 48 hours prior to the test. You should also look over the past tests/midterms and understand those fully. In addition, this information sheet provides some practice problems that are provided to help you study if you have finished all the homework questions. These practice questions do NOT form a model for the test. These questions are intended only to help you identify any gaps in your understanding. In the last 24 hours before the test, reread ALL the homework problems, skim through the lecture notes and past tests/midterms,
& go over these practice questions again.
• Try to keep your eyes on your own work during the test.
• All nontest items will have to be put by the wall or on my desk prior to the start of the test; this includes cell phones, which should be switched off.
• Any student who leaves the room during the test will not be allowed to continue the test. If you finish the test early, & wish to leave early, then please do so as quietly as possible. Please turn in your test and any additional paper (used or unused) when you leave.
• It is your responsibility to be on time.
• If you wish to know your grade early, turn in to me a STAMPED ADDRESSED envelope with the following listed on a piece of paper inside:
(a)
(b)
(c)
(d)
(e)

MATH 3330
Your full name written clearly
The last 4 digits of your ID number
Grade on Final =
Grade in course =

I will not give out any grades by e-mail.
Page 1 of 11

The following questions are for your practice for the test to see how I might phrase some questions
(especially multiple choice); they will not be graded and do NOT form a model for the test. Finish the homework assignments first, before working them.
PRACTICE QUESTIONS
1. Which of the following systems of equations is useful for determining the function(s) f (t) = a + bt + ct2 whose graph passes through the points (0, 2), (1, 8), (−1, 18)? a = −2 a + 4b = −8 a − b + c = −18

a= 2
(b) a + b + c = 8 a − b + c = 18

(c)

a = −2
(d) a + b + c = −8 a − b + c = −18

b= 2 a+c= 8
(e)
a − b + c = 18

a+b= 2
(f) a + c = 8 b − c = 18.

(a)

a= 2 a + 4b = 8 a − b + c = 18

2. Which of the following systems of equations is useful for determining the function(s) f (t) = a cos(2t)+b sin(2t) for all t ∈ R?

that satisfy the equation

(a)

4a − b = 0 a + 4b = −5

(b)

4a + b = 0 a − 4b = −5

(e)

a+b= 0 a + 2b = −5

(f)

a + 4b = 0 a + 4b = −5

d2 f df +2 +3f (t) = 5 cos(2t) dt2 dt a + 4b = 0 a + b = −5

a−b= 0 a − 2b = −5.

(c)

(d)

3. Write down the coefficient matrix and the augmented matrix (stating which is which) for the x1 − x2 − x3 = 0
2x1 − 3x2 = −1 system x2 − 2x3 = 1.

4. Which

1
0
(a)
0

0
(f) 0
0

of the following matrices is not in row-echelon form?









0010
2488
3456
0120
000
0 1 2 (b) 0 0 1 3 (c) 0 6 7 0 (d) 0 0 9 10 (e) 0 0 0 0
0000
0234
0081
0001
000



001
1001
0 0 0 (g) 0 0 0 0.
000
0000



4 −6
5. If A = −8 12 , then which of the following is a nontrivial solution of the equation Ax = 0?
6 −9
(a)

1
2

(b)

3
2

(c)

2
3

(d)

2
1

(e)

0
0

(f)

Page 2 of 11

−1
2

(g)

1
0

(h)

0
.
1


1
3 −5
4 −8, then which of the following best describes the set of solutions to Ax = 0
6. If A =  1
−3 −7 9 geometrically? 

(a) The zero vector in R2
(e) a plane in R4

(b) The zero vector in R3

(f) R3

(g) R4

7. The system of linear equations

(c) a line in R3

(d) a plane in R3

(h) 3 points in R3 .

x1 + 4x2 + 3x3 = 1
2x2 + x3 = 1 x2 + x3 = 1

has

(a) no solution (b) exactly 1 solution (c) infinitely many solutions (d) exactly 2 solutions.


1132
8. If 1 2 4 3 is the augmented matrix for a system of linear equations, then for which value(s)
135k
of k ∈ R is the system consistent?

(a) k = 1

(b) k = 2

(c) k = 4

(e) all k ∈ R except k = 2

(d) all k ∈ R except k = 1

(f) all k ∈ R except k = 4.

9. Justify or disprove: if a given vector b is a linear combination of the columns of a matrix A, then the equation Ax = b is consistent.

0303
1 2 0 4 is the augmented matrix for a system of linear equations, then the solution set
0015
system in vector form is:






3
3
−3
3
−3
−3
4
4
−2
4
−2
−2
(a) x =   + x3  
(b) x =   + x3  
(c) x =   + x3  
5
0
1
5
1
0
0
0
5
0
0
1


1
0
10. If
0
of the

(d) there are no solutions (e) not enough information given to answer question.

11. Which of the following statements is always true?
(a) A homogeneous system of linear equations is always consistent.
(b) A homogeneous system of linear equations is never consistent.
(c) A homogeneous system of linear equations always has nontrivial solutions.
(d) If A is a 3 × 2 matrix, then A must have a column without a pivot.

12. Compute

(d)

00
00

−1 0 1
2 34

(e)

12
.
34

−1 2 −3
−1 6 −5

(a)

227
−2 4 10



−1 −1
6
(f)  2
−3 −5
Page 3 of 11

(b)

1 −2
60

(g) cannot be multiplied.



−2 −2
4
(c)  2
7 10


0
2x1 − x2 + x3 = 0
0
  is a particular solution to the system
3x1 + 2x2 − x3 + x4 = 1
13. Noting that the vector v =  
0
5x1 + 8x2 − 5x3 + 3x4 = 3,
1
find all solutions to the system.






0
1
0
1
0
1
1
0
1
0
1
0
(a) v + s   + t   (b) v + s   + t   (c) v + s   + t  
1
−2
1
2
1
−2
1
−5
−1
−5
−1
−5
(d) none of the above.




10
2
1 2 12
5 ,
14. Given that the reduced row-echelon form of 0 1 5  is 0 1
0 0 k−7
11k



2
1
12
 5  a linear combination of 0 and 1? for which value(s) of k is
1
1 k (a) all k ∈ R except 7

(b) k = 0 only

(c) k = 7 only

(d) k = 0 & k = 7 only

(e) all k ∈ R except 0 & 7 (f) all k ∈ R (g) there is no such k

x1 x2 15. Define T : R2 → R2 by T

=

 
 x1
 

 x2

2


(h) not enough information given.

if x1 = 0

x1








0 x2 if x1 = 0.

Show that there exist u, v ∈ R2 such that T (u + v ) = T (u) + T (v ). (Hint: consider

T

T

0
1

(a)

13
7

−1
6

=

, find T

−7
43

(b)

(c)

5
−3

1
0

(e)

16. If T : R2 → R2 is a linear transformation such that

0
0

2
5

=

1
1
.) and 0
2

and

.

−13
−7

(d)

10
15

17. If T : R2 → R3 is a linear transformation such that T (u) =

(f)

2
.
23

1
2

and T (v ) =

−3
, find
2

T (2u − 3v ).
(a)

−2
4

(b)

4
2

(c)

11
−2

(d)

1
1

(e)

3
0

(f) not enough information is given.

Page 4 of 11

x1 x2 18. Find the matrix of the function T : R2 → R, where T
(a) x1 x2
2

(b) 0 1

00
0 x2
2

(c)

= x2 .
2
(e) x2
2

(d) 0 x2

(f) no such matrix.

−0.6 of length 1. Find the matrix of the
0.8

19. Suppose a line ℓ in R2 contains the vector u =
(orthogonal) projection map onto ℓ.

0.36 −0.48
−0.48 0.64

(a)

0.64 −0.48
−0.48 0.36

(b)

0.48 −0.64
−0.64 0.36

(c)

(d)

0.48 −0.36
−0.36 0.64

(e)

0.64 −0.36
−0.36 0.48

(f) no such matrix.

20. Find the shear matrix that transforms
(a)

10
51

21. If A =

(b)

50
05

(c)

50
11

3
3
. into 16
1
(d)

15
01

(e)

11
05

(f) no such shear.

k2
, then for which value(s) of k is A not invertible?
−3 4

(a) 2

(b)

3
2

(c) −2

(d) − 3
2

(e) 0

(f) 0 & 2.


1 23
22. If A = −1 2 0, then which of the following is the second row of A−1 ?
2 14


(a) 1 1 1
(b) 0 0 1
(c) 4 − 2 − 3
(d) 1 2 − 3
−1
(g) 0 − 2 1
(h) 0 2 1
(i) A does not exist.

(e) 2 3 4

(f) 4 5 − 6




7
5
1
2, v2 = 6, v3 = 10. Describe span(v1 , v2 , v3 ) = Rv1 + Rv2 + Rv3 .
23. Let v1 =
9
1
4
(a) a point
(b) a line
(c) a plane
(d) R3
(f) exactly 3 vectors emerging from the origin in R3 .

(e) exactly 3 points


1
3 −5
4 −8, then which of the following best describes the kernel of A geometrically?
24. If A =  1
−3 −7 9


(a) The zero vector in R2

(d) a line in R3

(b) The zero vector in R3

(e) a plane in R3

(f) a plane in R9

Page 5 of 11

(c) 3 points in R3
(g) R3

(h) R4 .


1
3 −5
4 −8, then which of the following best describes the image of A geometrically?
25. If A =  1
−3 −7 9


(a) The zero vector in R2

(d) a line in R3

(b) The zero vector in R3

(e) a plane in R3

(c) 3 points in R3

(f) a plane in R9

(g) R3

(h) R4 .

 

 x1

x2  : x1 , x2 ∈ R
26. The set U = is not a subspace of R3 because


5

0
(a) U is not a subset of R3 (b) U is empty (c) U contains 0 (d) U contains the origin
5
(e) U does not contain the origin (f) trick question: U is a subspace of R3 .





2
5
1
9
 1 , v2 = 1, v3 =  2 . Given that −7 is in the
27. Let v1 =
1
−1
33
4


129
kernel of the matrix A =  1 1 2 , find a nontrivial relation among v1 , v2 and v3 .
−1 4 33


Show some work; write clearly.

28. Which of the following sets of vectors is linearly independent?
     
     
      
5
6
3
1
0
2
1
2
1
5 , 6 , 1
−1 , −2 , 0
 1  , 3 , 0
(c)
(b)
(a)






3
6
3
5
1
3
0
4
−1
     
   
2
4
2
6
1
7 , 1 , 0
2 , 4
(f) none of these.
(e)
(d)




0
0
8
6
3

29. Which of the sets in the previous question is a basis for R3 ? (a) (b) (c) (d) (e) (f) none.
30. Define (i) subspace of Rn (ii) span of m vectors in Rn (iii) kernel of a linear transformation
(iv) image of a linear transformation (v) basis of a subspace (vi) dimension of a subspace (vii) the coordinate vector of a vector with respect to a basis B.

 

d



  c  : a, b, c, d ∈ R and a + d = b + c is a subspace of R4 . Find the dimension
31. The set W =  

b



 a of W .
(a) 7

(b) 6

(c) 5

(d) 4

(e) 3

(f) 2

Page 6 of 11

(g) 1

(h) 0.

32. You are given a matrix A and its reduced row-echelon form:




1 0 3 0 0 −1 4
1 0 3 0 0 −1 4
1 1 3 0 0 2 3
0 1 0 0 0 3 −1




1 1 3 1 0 1 7
0 0 0 1 0 −1 4 


A=
rref(A) = 
1 1 3 1 1 4 6
0 0 0 0 1 3 −1.




1 1 3 1 1 4 6
0 0 0 0 0 0
0
11311 4 6
00000 0
0
The matrix A represents a linear transformation T where T (x) = Ax and....
(a) T : R13 → R6

(b) T : R6 → R13

(e) T : R6 → R7

(c) T : R7 → R6

(f) T : R7 → R13

(g) T : R13 → R7

(d) T : R42 → R42
(h) not enough information.

33. Find the dimension of the image of the matrix A given in Question 32.
(a) 6

(b) 5

(c) 4

(d) 3

(e) 2

(f) 1

(g) 0

(h) not enough information.

34. Find a basis for the image of the matrix A given in Question 32.
(a) Rows 1, 2, 4 of rref(A)

(b) rows 1-4 of rref(A)

(c) rows 1, 2, 4, 5 of A

(d) all the rows of A

(e) columns 1, 2, 4, 5 of rref(A)

(f) all the columns of rref(A)

(g) columns 1, 2, 4, 5 of A

(h) all the columns of A.

35. Using the matrix A from Question 32 and a vector x, how many free variables has the homogeneous system Ax = 0?
(a) 7

(b) 6

(c) 5

(d) 4

(e) 3

(f) 2

(g) 1

(h) 0.

36. Find the dimension of the kernel of the matrix A given in Question 32.
(a) 7
(b) 6
(c) 5
(d) 4
(e) 3
(f) 2
(g) 1
(h) 0.
37. Find a basis for the kernel of the matrix A given in Question 32; show all work.
38. Let A =
(a)

1
2

2
5
−3 10
.
, v2 =
, v1 =
1
3
−3 8
(b)

4
2

(c)

0
2

(d)

5
3

Using B = {v1 , v2 } as a basis for R2 , find [Av2 ]B .
(e)

25
13

(f) not enough information.

39. Let T (x) = Ax for all x ∈ R2 , where A is the 2 × 2 matrix from Question 38. Find the matrix of T with respect to the basis B given in Question 38.
(a)

2 10
08

(f)

30
02

(b)
(g)

−3 10
−3 8
10
01

(c)
(h)

20
03

(d)

2 −3
.
0 −3

Page 7 of 11

−3 8
−3 10

(e)

8 −10
3 −3





aa
40. Find a basis for the real linear space V =  b b  ∈ R2×3 : a, b, c ∈ R .

 cc 
1
(a) 1

1

1
(e) 0

0


0
0 ,
0

0
0 ,
0


0
0
0

0
1
0





00
1
1 1
1
(b) 0 0 , 1 1 ,


00
00
1




0 0
0
0 1
0 , 0 0
(f) 0 0 ,


00
10
0


0
0
1

0
0
0



0 0
1
1 , 1 0

11
0





0 0
1
1 1
(d) 0
(c) 1 0 , 0 1



0
11
00

0
V has a basis,
0
(g)

but it is infinite.
1


0
0

1

0
0
1 , 0
0
0

41. Let A, B ∈ Rn×n . We define a map T : Rn×n → Rn×n by T (M ) = AM B for all M ∈ Rn×n .
Justify whether or not T is a linear transformation. Show all work.
42. Let T : P2 → R2 be given by mial f ; i.e., T (f ) =

4f (t)

t=1 df dt t=4

T (f ) =

4f (1)
, where f ′ denotes the derivative of the polynof ′ (4)

. You may assume that T is a linear transformation. Using the

basis {t2 , t, 1} for P2 and the basis

0
1
,
1
0

for R2 , find the matrix of T with respect to

these bases.
(a)

444
840

(b)

444
810





40
48
444
(c) 4 1 (d) 4 1 (e)
018
48
40



444
(f) 8 1 4.
001




0
0
1
0, e2 = 1 and e3 = 0, and let E = {e1 , e2 , e3 } be the standard basis of R3 ,
43. Let e1 =
1
0
0
and assume B = {−e2 , e1 , e1 + e3 } is another basis of R3 . Find the matrix which changes basis from B to E .








1
00
1 −1 0
10 0
101
(a) 0 0 1 (b) 0 0 −1 (c) 0 0 −1 (d) −1 0 1
0 −1 0
01
0
11 0
0 −1 0








0
10
0 −1 0
0 11
0 −1 0
(e) 1 0 0 (f) −1 0 0 (g) 1 0 −1 (h) −1 0 0.
0 −1 1
00
1
0 01
101
44. Which of the following sets of vectors form an orthonormal set?
 2     2 
  2     2 
1
√
−3 
−3 
0
3
3




 0   2  0 
0   0
(a)   ,  1  ,  
(b)   , −1 ,  
√
√
 √5
 √5




5
5
0
0
3

3

 2     √ 
5
0
3


3
 

(d)  0  , 1 ,  0 
√




2
0
− 35
3

3

3

 2   1   2 
√
−3 
3


5
 0  √   0 
1
(e)   ,  5  ,  
√

 √5

1
5
√
3

5

Page 8 of 11

3

  2     2 
−3 
0
3


0   0
(c)   , 1 ,  
√
 √5


5
0
3

(f) none of them.

3

45. Use the Gram-Schmidt orthogonalization procedure to produce an orthonormal basis for the sub

2
2
3
−2 and  0 . space of R spanned by
−1
0

 1 
 1   
 1  
 1   

1
√
 − √2
1
1
0

 √2
 √2

 √2
5








 1  1
1  
 1  2 
 1  
(a)  √2  ,  √5 
(c)  √  ,  1 
(d) − √  ,  √5 
(b) − √  ,  1 
 √ 
2
2



2











1
0
− √5
√3 
0
0
0
−1
−1
5


    
 1   2 
  √ 
  √ 
1
1
1
1
√
√


√
√
0

0





3
3
5
2
2
2










 1 √ 
 1 
 1  − √ 
 1
1.
(g) 0 , 1
(h) − √  ,  3 
(f) − √  ,  0 
(e) − √  ,  3 
2
2
2









1





1
1
1
0

0
0
√
− √5
0
−√ 
3

3

46. Which of the following matrices is orthogonal?
2

1


−3 2
3
3
2 −1 0
34
1
2
(b) − 3 2
(a)
(c) 0 0 1
3
3
4 −3
120
2
2
1
−3
3
3

2
2
1
3
3
1
1
√
√
20

1 2
2
2
(g) 1
(f) − 3 3 0
(e)
.
1
02
√
√
2
2
2
−1 0
3
3


1
2 −3
47. Find the determinant of  3 −1 1 .
−1 1 −2



1
2
1
2


(d) − 1
3
1
3

−1

0

1
3

2




1



(a) −15

(b) −1

(c) 1

(d) 0

(e) 2

(f) 3

(g) 5

(h) 7

(i) 9.

48. If A, B and C are n × n matrices such that det(A) = 5, det(B ) = 2, det(C ) = 4, find det(AB −1 C T ).
(a) −4

(b) −2

(c) 0

(d) 2

(e) 4

(f) 6

(g) 8

(h) 10

(i) 12.

49. If A is a 5 × 5 matrix, find det(2A).
(a) 128 det(A)
(f) −2 det(A)

(b) 32 det(A)
(g) −8 det(A)

(c) 8 det(A)
(d) 2 det(A)
(e) 0
(h) −32 det(A)
(i) −128 det(A).

50. Let A denote an n × n matrix. Give the definition of an eigenvalue λ of A.



1
11 2
51. Let A = 2 1 −3. The matrix A has an eigenvector v = −1 associated to an eigenvalue λ;
1
03 5 find λ. (Hint: the fastest way to do this question does not use any polynomials.)

(a) 0

(b) −1

(c) 1

(d) −2

(e) 2

(f) −3

Page 9 of 11

(g) 3

(h) −4

(i) 4.

52. A linear transformation T : R3 → R3 has an eigenvector v ∈ R3 associated to an eigenvalue λ = 3; find T 4 (v ).
(a) 243v

(b) 9v

(c) 81v

(d) 99v

53. Find the characteristic polynomial of
(a) 1 − x

(b) x2 + 4x + 1

(f) x2 − 4x + 5

(e) 3v

(g) not enough information.

31
.
21

(c) x2 + 4x − 1

(g) x2 + 4x − 5

(f) 27v

(d) x2 − 4x + 1

(h) x2 − 4x − 5

(e) x2 − 4x − 1

(i) x2 + 4x + 5.



200
54. Find all the eigenvalues of A = 0 −3 5.
0 −6 8

(a) 1, 2 & 3 only
(b) 2 & 3 only
(c) 2, −3 & 8 only
(d) 1 & 3 only
(f) 1 only
(g) 2 only
(h) 3 only
(i) A has no eigenvalues in R.

(e) 1 & 2 only


0 1 −3
55. The matrix A = −3 4 −3 has eigenvalue 3. Find the eigenspace of the eigenvalue 3. Show
003
all work and write clearly.


56. If A = SDS −1 , where S =

10
11

and D =

(a)

2p − 3p 3p
2p
0

(b)

2p 2p − 3p
0
3p

(e)

3p
0
p p 2 − 3 2p

(f)

20 p , find A for any positive integer p.
03

2p
0
2 − 3p 3p

0
2p
p p 3 2 − 3p

3p
0
. p p
3 − 2 2p

(c)

p

(d)

57. A certain 3 × 3 matrix A has eigenvalue 5 with algebraic multiplicity 3. Which one of the following matrices is similar to A?










500
500
510
510
500
not enough
(a)0 5 0 (b)0 5 0 (c)0 5 1 (d)1 5 0 (e)1 5 0 (f) information. 015
005
005
005
005



0
1
1
58. A 3 × 3 matrix A has eigenspaces E1 = R  1  + R 2 and E2 = R 1.
−1
1
0
−1 following matrices S does NOT make S AS diagonal?









11
01 1
011
1 10
1 01
(a) 1 2 1 (b) 1 1 2 (c)2 1 1 (d)2 1 1  (e)1 1
0 −1
1 0 −1
1 −1 0
−1 0 1
−1 1 0






20 1
10 1
2
11
 (g)1 2 1  (h)2 2 1 .
4
11
(f)
0 1 −1
0 1 −1
−5 −1 0


Page 10 of 11

Which of the


0
2
1



0
1
1 and E2 = R 1.
59. A certain 3 × 3 matrix A has only the eigenspaces E1 = R
1
1 invertible matrix S such that S −1 AS is diagonal.











100
100
010
001
010
21
1 2 1 (b)1 1 2 (c)2 1 1 (d)2 1 1 (e)1 1 2 (f)4 1
(a)
111
111
111
111
111
21




001
001
(g)1 2 1 (h)2 2 1 (i) no such matrix.
211
111

Find an


0
1
1






1
1
0
−1 and E1 = R 0 + R 1. Find
60. A symmetric 3 × 3 matrix A has eigenspaces E4 = R
0
0
1
an orthogonal matrix S such that S −1 AS is diagonal.
√
1
1


1

1
0 √2
√
− √2 0
2
1 01
2
1
1
0
1
(a)− √2 0 √2 
(c)−1 0 1
(b) 0
1
1
√
√
0 10
0
2
2
0
10


1 −1 0
(d)0 0 1
110

(e) not enough information.

Remember: most of the test will be based on the homework questions, so finish those questions and look over your solutions to them before the test. Also look over the previous tests and midterms.
Reread Page 1 above to see what to bring to the test.

Page 11 of 11