Description

6 questions already completed! Please complete the following which is due in 6 hours. Please solve with full details and all work shown.

1 attachmentsSlide 1 of 1attachment_1attachment_1

Unformatted Attachment Preview

Name:

Linear Algebra

Date:

Exam 2 – Take Home

Submission Protocols The completed quiz should be uploaded – as a single PDF file – to the MyMathLab Document Sharing forum. Please see the submission protocols sent to you via email. However, I summarize the document

naming protocols below:

Please put your “last name”, “first name”,“quiz number”,“submission number”,“course” in the file name –

separated by underscores (without the quotes or commas). For example, the filename for the first submission

for student John Doe should be: Doe John Exam2 P2 V1 2318 .

Quality of Submitted Work

Before you submit your quiz, you must first examine the PDF file to make sure that your work is legible and that there are no

large black borders surrounding the images (this will waste too much toner when printing). In other words, make sure that

your work is legible in the scanned image AND please try to ensure that I will not be wasting printer toner when printing.

Instructions Show all work and pertinent calculations (particularly when performing elementary row operations). An answer is not considered to be correct without complete and correct supporting work. In other words, for each

question, to receive full credit you must show all work. Explain your reasoning fully and clearly. I expect your solutions

to be well-written, neat, and organized. Do not turn in rough drafts. What you turn in should be the “polished” version

of potentially several drafts. Once again, show all of your work and justify your solutions fully. In addition, the simple

operational rules for this quiz are:

A. Calculators or computer software solutions will not be accepted.

Students are expected to present their own analytical solutions.

B. You may refer to theorems in the book (unless the question specifically states

otherwise), but only cite theorems that are from the appropriate sections in our

textbook; i.e., those that come from the relevant sections of Chapters 1, 2 and 3.

Nota Bene:

To receive full credit on graded work, it is important that you write each step of the row reduction process correctly. Furthermore, when completing a proof or justifying your work it is important that you explicitly state your conclusion and reasons.

When performing elementary row operations, please use a style that is similar to that outlined in the example.

Example. Show the that the linear system associated with

Solution.

1

2 −1

4

7

2

6 −7

−2R1 + R2 → R2

1R1 + R3 → R3

−1 −1 −2 −1

∼

4 1R3 + R1 → R1

1 2 −1

3

1 −3

3R3 + R2 → R2

0 0

1 −7

0

∼

∼

1 2 −1

4

2 −5 −1

3 R3 ↔ R2

0 1 −3

1 2 0 −3 −2R2 + R1 → R1

1 0 −18

0 0 1 −7

0

0

∼

4

1

2 −1

2

6 −7

7

−1 −1 −2 −1

is consistent.

1 2 −1

4

3

1 −3

0 2 −5 −1 −2R2 + R3 → R3

0

∼

33

1 0 0

1 0 −18

0 0 1 −7

0

.

Since the there is no pivot position in the rightmost column, we know the associated linear system is

consistent. Finally, the solution to the system is (33, −18, −7).

This portion (i.e., Part 2) of the entire quiz is worth 40 points.

Do not submit this cover page.

–Created/Revised by Mr. Sever 3/5/20

Page 0 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

1. (6 pts.) Let A =

a) (A + B)2

b) (AB T )

1 2 2

−2

1 −1

3 1 2

5

−3

4

,

B

=

. Compute:

−1 2 4

−2

1

6

T

c) A2 − 3A + 4I

–Created/Revised by Mr. Sever 3/5/20

Page 1 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

2. (8 pts.) Let A =

1 1 1

1 2 3

. Write A as the product of elementary matrices.

1 4 5

–Created/Revised by Mr. Sever 3/5/20

Page 2 of 9

Linear Algebra

Name:

Date:

Exam 2 – Take Home

3. (9 pts.) Use the row reduction algorithm to find the inverse of A =

–Created/Revised by Mr. Sever 3/5/20

1

1

2

1

1 1

5

1 2

0

3 −1 1

0 1

2

.

Page 3 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

4. (4 pts.) Determine which of the following matrices are elementary.

0 1 0

E1 = 1 0 0 ,

0 0 1

0 0 1

E7 = 0 1 0 ,

1 0 0

E2 =

”

#

1 −1

,

0

1

1 0 0

E8 = 0 0 0 ,

0 0 1

1 0 0

E3 = 0 2 0 ,

0 0 1

E9 =

”

√

#

π 0

,

0 1

E4 =

E10

#

”

0 0

,

0 0

”

#

0 1

,

=

1 0

1 0 −2

E5 = 0 1 0 ,

0 0

1

E11

”

#

0 2

,

=

1 0

E12

E6 =

”

#

1 0

,

5 1

1 0 0

= 0 1 0 .

0 0 1

In the list below, circle the matrices that are elementary:

E1

E2

E3

E4

E5

E6

E7

E8

E9

E10

E11

E12

5. (6 pts.) Construct a 4 × 4 matrix A and a vector b such that b is not in Col(A) and rank(A) = 2.

Show that your matrix has the desired property.

–Created/Revised by Mr. Sever 3/5/20

Page 4 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

6. (6 pts.) Suppose that A, B, and X are n × n matrices with A invertible.

Solve XAT

T

+ 3AB 2 = AIn A for A.

7. (4 pts.) Given that det(AT ) =

8

for a 3 × 3 matrix A. Find the determinant of A.

5

1 4

3

8. (6 pts.) Find the determinant of A = 2 1 2

.

4 3 −2

–Created/Revised by Mr. Sever 3/5/20

Page 5 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

9. (10 pts.) The reduced echelon form of A =

3

0

4

1 −2 −5

−1

0

4

1 −2 −1

1

0

7

0

2 −2

is

0

0

3

0

2

2

0

1

0

0

0

0

1

0 −1 −1

.

1

0

1

0

0

0

a) Find a basis for the null space of A.

b) Find a basis for the column space of A.

c) State the dimension of the null space of A. Answer:

d) State the dimension of the column space of A. Answer:

–Created/Revised by Mr. Sever 3/5/20

Page 6 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

10. (4 pts.) Let B =

11. (8 pts.) Let B =

(”

−1

−1

−3

7

1

, 5

−4

−6

# ”

,

−2

1

#)

2

be an ordered basis of R . Find x ∈ R such that [x]B =

”

#

6

.

4

−3

be an ordered basis of H = Span{B}. Find 1 .

−4 B

12. (4 pts.) Find all values of h such that A =-

–Created/Revised by Mr. Sever 3/5/20

2

”

2h 4h

2 h2

#

is singular.

Page 7 of 9

Linear Algebra

Name:

Date:

Exam 2 – Take Home

13. (12 pts.) Use row reduction to find the determinant of A where A =

–Created/Revised by Mr. Sever 3/5/20

1

2

3

1

4

1

2

0

−3

1

−3 −1

.

−3

1

−1

1

Page 8 of 9

Name:

Linear Algebra

Date:

Exam 2 – Take Home

a

b

14. (13 pts.) Let H =

:a+b+c=0 .

c

a) Show that H is a subspace of R3 by finding a matrix A such that Nul(A) = H.

b) Using your result in part a), find a basis for H.

–Created/Revised by Mr. Sever 3/5/20

Page 9 of 9

Purchase answer to see full

attachment

Tags:

math equations

Algebraic equations

math test

algebra functions

linear algebra

User generated content is uploaded by users for the purposes of learning and should be used following Studypool’s honor code & terms of service.

## Reviews, comments, and love from our customers and community:

This page is having a slideshow that uses Javascript. Your browser either doesn't support Javascript or you have it turned off. To see this page as it is meant to appear please use a Javascript enabled browser.