Linear Algebra HW

Hussam Malibari 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

 

−7 1 −1 2

  and

 

7 −2 7 −6

 ?

• A.  

35 −7 17 −18

 

• B.  

119 −19 62 −72

 

• C.  

18 −36 46 −36

 

• D.  

−7 −7 47 −30

 

2. (1 point) Which of the following sets of vectors are lin- early independent?

• A. {[

6 9

] ,

[ −6 −9

]} • B.

    −58

0

 ,   2−7

0

 ,   4−9

0

   

• C. {[

−2 5

] ,

[ 3 −8

] ,

[ 6 9

]} • D.

{[ −7 4

]} • E.

    1−8

4

 ,   09

−6

 ,   52

3

   

• F. {[

0 0

] ,

[ −3 8

]}

3. (1 point) Let A =

  20

−2

 , B =

  21

−3

 , and C =

  −40

7

 .

? 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.

4. (1 point) Let A =

 

−3 12 −7 6

 , B =

 

0 2 −1 −1

 , C =

 

−1 4 −1 2

 ,

and D =

 

−2 10 −5 3

 .

? 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.

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5. (1 point) Find a basis for the subspace of R4 spanned by the following vectors.

 −1 0 0 0

  ,  

−2 −1 1 −2

  ,  

−5 −2 2 −4

  ,  

−1 2 1 −1

 

Answer:

6. (1 point) Find a basis for the subspace of R4 consisiting of all vectors of the form 

 x1

9×1 + x2 8×1 − 8×2 −4×1 − 3×2

 

Answer:

7. (1 point) Find a basis for the subspace of R3 consisting of

all vectors

  x1x2

x3

  such that −6×1 − 7×2 − 6×3 = 0.

Hint: Notice that this single equation counts as a system of linear equations; find and describe the solutions.

Answer:

8. (1 point) Consider the ordered basis B of R2 consisting of

the vectors [

3 1

] and

[ −2 3

] (in that order). Find the vector~x

in R2 whose coordinates with respect to the basis B are [

−4 1

] .

~x = [ ]

9. (1 point) The set B = {[

−5 −2

] ,

[ 15 9

]} is a basis for

R2. Find the coordinates of the vector ~x = [

30 15

] with respect

to the basis B:

[~x]B = [ ]

10. (1 point) Suppose that A is a 7 × 5 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.

11. (1 point) Let A =

 

1 0 0 −4 −2 4 0 0 −16 −8 0 1 0 1 −3 0 1 −2 −9 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 −4 −2 0 1 0 1 −3 0 0 1 5 −3 0 0 0 0 0

 .)

Row Space basis: Column Space basis: Null Space basis: Rank: Nullity:

12. (1 point) Let A =

 

1 1 −1 0 0 0 4 4 −4 0 0 0 0 0 0 0 −2 −3 0 0 0 0 1 3 0 0 0 −1 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 1 −1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0

 .)

Row Space basis: Column Space basis: Null Space basis: Rank: Nullity:

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Hussam Malibari 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 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) Find the characteristic polynomial of the matrix[ 10 −6 1 10

] . (Use x instead of λ.)

p(x) = .

2. (1 point) Find the characteristic polynomial of the matrix  −1 −1 00 1 −3

−5 5 0

 . (Use x instead of λ.)

p(x) = .

3. (1 point) The eigenvalues of

  8 −16 160−2 4 −88

0 0 −12

  are

. (Enter your answer as a list of numbers; for example, 1, 2, 3.)

4. (1 point) The eigenvalues of

 

−3 −4 4 −1 0 3 −5 1 0 0 4 3 0 0 0 −2

  are

. (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.)

5. (1 point) 3 The matrix A =

 

−3 0 0 0 0 −3 −3 −3 3 0 2 2 −3 0 −2 −2

  has

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 .

6. (1 point) The matrix A =

  0 0 05 5 0

5 5 0

  has two real

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 .

7. (1 point) Let A = [

−2 2 0 −1

] . Find an invertible matrix

P and a diagonal matrix D such that A = PDP−1.

P = [ ]

, D = [ ]

,

8. (1 point) The matrix C =

  37 0 −8412 −5 −24

18 0 −41

  has two

distinct eigenvalues, λ1 < λ2: λ1 = has multiplicity . The dimension of the corresponding eigenspace is . λ2 = has multiplicity . The dimension of the corresponding eigenspace is . Is the matrix C diagonalizable? (enter YES or NO)

9. (1 point) If n is a positive integer, then [

2 −24 −4 −2

]n is[ ]

(Hint: Diagonalize the matrix [

2 −24 −4 −2

] first. Note that

your answer will be a formula that involves n. Be careful with parentheses.)

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