Salem Almarar Heckman MAT 242 Spring 2017 Assignment Chapter 4 due 04/04/2017 at 11:59pm MST In some of the problems in this chapter, you will be asked to enter a basis for a subspace. You should do this by placing the entries of each vector inside of brackets, and giving a list of these vectors, separated by commas. For instance, if your basis is    12 3  ,   11 1    , then you would enter [1,2,3],[1,1,1] into the answer blank. 1. (1 point) Which of these vectors can be written as a linear combination of   −3 −5 −2 2   and   1 5 5 −7  ? • A.   14 50 44 −60   • B.   −17 −20 −20 −6   • C.   −45 −41 −29 −89   • D.   29 75 54 −70   Answer(s) submitted: • ( A, C ) (incorrect) 2. (1 point) Which of the following sets of vectors are lin- early independent? • A.     −6−2 −3  ,   −86 7  ,   −28 10     • B. {[ 2 4 ] , [ −2 −4 ]} • C. {[ 0 0 ] , [ −6 −7 ]} • D.     8−7 0  ,   3−5 0  ,   9−4 0     • E. {[ −5 9 ]} • F. {[ −1 3 ] , [ 8 −1 ]} Answer(s) submitted: • (incorrect) 3. (1 point) Let A =   −10 −4  , B =   −72 −30  , and C =   −22 −10  . ? 1. Determine whether or not the three vectors listed above are linearly independent or linearly dependent. 2. If they are linearly dependent, find a non-trivial linear combi- nation of A,B,C that adds up to~0. Otherwise, if the vectors are linearly independent, enter 0’s for the coefficients. A+ B+ C = 0. Answer(s) submitted: • Linearly_Independent • 5 (incorrect) 1 4. (1 point) Let A =   12 −28 −15 14  , B =   8 −19 −12 6  , C =   2 −5 −3 2  , and D =   −4 9 5 −4  . ? 1. Determine whether or not the four vectors listed above are linearly independent or linearly dependent. 2. If they are linearly dependent, find a non-trivial linear com- bination which adds up to the zero vector. Otherwise, if the vectors are linearly independent, enter 0’s for the coefficients. A+ B+ C+ D = 0. Answer(s) submitted: • • (incorrect) 5. (1 point) Find a basis for the subspace of R4 spanned by the following vectors.  1 2 −2 1   ,   −4 −8 8 −4   ,   −3 −6 6 −3   ,   2 −1 −1 2   Answer: Answer(s) submitted: • (incorrect) 6. (1 point) Find a basis for the subspace of R4 consisiting of all vectors of the form   x1 −2x1 + x2 −9x1 − 9x2 −5x1 + 2x2   Answer: Answer(s) submitted: • (incorrect) 7. (1 point) Find a basis for the subspace of R3 consisting of all vectors   x1x2 x3   such that 6x1 − 7x2 + 4x3 = 0. Hint: Notice that this single equation counts as a system of linear equations; find and describe the solutions. Answer: Answer(s) submitted: • (incorrect) 8. (1 point) Consider the ordered basis B of R2 consisting of the vectors [ 2 −3 ] and [ −5 5 ] (in that order). Find the vector ~x in R2 whose coordinates with respect.

1. Salem Almarar Heckman MAT 242 Spring 2017
Assignment Chapter 4 due 04/04/2017 at 11:59pm MST
In some of the problems in this chapter, you will be asked to
enter a basis for a subspace. You should do this by placing the
entries
of each vector inside of brackets, and giving a list of these
3
1
then you would enter [1,2,3],[1,1,1] into the answer blank.
1. (1 point) Which of these vectors can be written as a linear
combination of

2. −3
−5
−2
2
1
5
5
−7
14
50
44
−60

3. −17
−20
−20
−6
−45
−41
−29
−89
29
75
54
−70
Answer(s) submitted:
• ( A, C )
(incorrect)

4. 2. (1 point) Which of the following sets of vectors are lin-
early independent?
• A.
−3
7
10
• B.
{[
2

5. 4
]
,
[
−2
−4
]}
• C.
{[
0
0
]
,
[
−6
−7
]}
• D.
0

6. 3−5
0
0
• E.
{[
−5
9
]}
• F.
{[
−1
3
]
,
[
8
−1

7. ]}
Answer(s) submitted:
•
(incorrect)
3. (1 point) Let A =
−4
−30
C =
−10
? 1. Determine whether or not the three vectors listed above

8. are linearly independent or linearly dependent.
2. If they are linearly dependent, find a non-trivial linear
combi-
nation of A,B,C that adds up to~0. Otherwise, if the vectors are
linearly independent, enter 0’s for the coefficients.
A+ B+ C = 0.
Answer(s) submitted:
• Linearly_Independent
• 5
(incorrect)
1
4. (1 point) Let A =
12
−28
−15
14
8

9. −19
−12
6
2
−5
−3
2
−4
9
5
−4
? 1. Determine whether or not the four vectors listed above
are linearly independent or linearly dependent.
2. If they are linearly dependent, find a non-trivial linear com-
bination which adds up to the zero vector. Otherwise, if the
vectors are linearly independent, enter 0’s for the coefficients.

10. A+ B+ C+ D = 0.
Answer(s) submitted:
•
•
(incorrect)
5. (1 point) Find a basis for the subspace of R4 spanned by
1
2
−2
1
−4
−8
8
−4
−3
−6
6

11. −3
2
−1
−1
2
Answer:
Answer(s) submitted:
•
(incorrect)
6. (1 point) Find a basis for the subspace of R4 consisiting of
x1
−2x1 + x2
−9x1 − 9x2
−5x1 + 2x2
Answer:
Answer(s) submitted:

12. •
(incorrect)
7. (1 point) Find a basis for the subspace of R3 consisting of
all vectors
x3
Hint: Notice that this single equation counts as a system
of linear equations; find and describe the solutions.
Answer:
Answer(s) submitted:
•
(incorrect)
8. (1 point) Consider the ordered basis B of R2 consisting
of the vectors
[
2
−3
]
and

13. [
−5
5
]
(in that order). Find the
vector ~x in R2 whose coordinates with respect to the basis B
are[
−6
−4
]
.
~x =
[ ]
Answer(s) submitted:
•
(incorrect)
9. (1 point) The set B =
{[
1
−5
]
,
[
3
−6

14. ]}
is a basis for
R2. Find the coordinates of the vector~x =
[
−1
5
]
with respect
to the basis B:
[~x]B =
[ ]
Answer(s) submitted:
•
(incorrect)
10. (1 point) Suppose that A is a 3 × 2 matrix.
(a) A vector in the null space of A has entries in it.
(b) A vector in the row space of A has entries in it.
(c) A vector in the column space of A has entries in it.
Answer(s) submitted:
•
•
•
(incorrect)

15. 11. (1 point) Let A =
1 0 0 0 5
2 0 0 0 10
0 0 −2 4 −8
0 −2 −2 8 −6
a basis for the row space of A, a basis for the column space
of A, a basis for the null space of A, the rank of A, and the
1 0 0 0 5
0 1 0 −2 −1
0 0 1 −2 4
0 0 0 0 0
Row Space basis:
Column Space basis:
Null Space basis:
Rank:
2
Nullity:

16. Answer(s) submitted:
•
•
•
•
•
(incorrect)
12. (1 point) Let A =
1 0 0 3 1 0
5 0 0 15 5 0
0 3 −3 −6 −3 1
0 0 0 0 0 1
0 −3 3 6 3 −3
Find a basis for the row space of A, a basis for the column
space
of A, a basis for the null space of A, the rank of A, and the
nullity of A. (Note that the reduced row echelon form of A is
1 0 0 3 1 0
0 1 −1 −2 −1 0
0 0 0 0 0 1

17. 0 0 0 0 0 0
0 0 0 0 0 0
Row Space basis:
Column Space basis:
Null Space basis:
Rank:
Nullity:
Answer(s) submitted:
•
•
•
•
•
(incorrect)
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Salem Almarar Heckman MAT 242 Spring 2017
Assignment Chapter 6 due 04/04/2017 at 11:59pm MST
In some of the problems in this chapter, you will be asked to
enter a basis for a subspace. You should do this by placing the
entries
of each vector inside of brackets, and giving a list of these

18. 3
1
then you would enter [1,2,3],[1,1,1] into the answer blank.
1. (1 point) Find the characteristic polynomial of the matrix[
8 −2
−2 9
]
. (Use x instead of λ.)
p(x) = .
Answer(s) submitted:
• (8-x)(9-x)-(-2)(-2)
(correct)

19. −5 −3 0
p(x) = .
Answer(s) submitted:
• (2-x)*(-(1)*(-5))-(-1)*(-(1)(-5))
(incorrect)
3. (1 point) The eigenvalues of
0 −1 3
.
(Enter your answer as a list of numbers; for example, 1, 2, 3.)
Answer(s) submitted:
• 0, 2, 4
(correct)
4. (1 point) The eigenvalues of

20. −1 −8 −9 6
0 0 3 −9
0 0 0 −8
0 0 0 8
.
(Enter your answer as a list of numbers; for example, 1, 2, 3,
3. If an eigenvalue is repeated, then it should be listed as many
times as appropriate.)
Answer(s) submitted:
• 0, 8, 1
(incorrect)
5. (1 point) 1 The matrix A =
1 0 0 0
−1 −2 3 −1
0 −2 2 0
0 −2 1 1
two distinct eigenvalues λ1 < λ2. Find the eigenvalues and a
basis for each eigenspace.
λ1 = , whose eigenspace has a basis of .
λ2 = , whose eigenspace has a basis of .

21. Answer(s) submitted:
•
•
•
•
(incorrect)
6. (1 point) The matrix A =
−2 −2 −2
eigenvalues, one of multiplicity 1 and one of multiplicity 2.
Find
the eigenvalues and a basis of each eigenspace.
λ1 = has multiplicity 1, with a basis of .
λ2 = has multiplicity 2, with a basis of .
Answer(s) submitted:
•
•
•
•
(incorrect)
7. (1 point) Let A =

22. [
1 2
0 0
]
. Find an invertible matrix P
and a diagonal matrix D such that A = PDP−1.
P =
[ ]
, D =
[ ]
,
Answer(s) submitted:
•
(incorrect)
8. (1 point) The matrix C =
−1 1 1
-
tinct eigenvalues, λ1 < λ2:
λ1 = has multiplicity .
The dimension of the corresponding eigenspace is .

23. λ2 = has multiplicity .
The dimension of the corresponding eigenspace is .
Is the matrix C diagonalizable? (enter YES or NO)
Answer(s) submitted:
•
•
•
1
•
•
•
•
(correct)
9. (1 point) If n is a positive integer, then
[
16 −48
4 −16
]n
is[ ]
(Hint: Diagonalize the matrix
[
16 −48
4 −16

24. ]
first. Note that
your answer will be a formula that involves n. Be careful with
parentheses.)
Answer(s) submitted:
•
•
•
•
(incorrect)
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