Description

qeqweqwrasc sadwqedqeqwewqkjfqwpehfpoqwjepoqwejoipfjwq[pofejp[owqejfp[owqejf[pwqekfp[

1 attachmentsSlide 1 of 1attachment_1attachment_1

Unformatted Attachment Preview

Zhenghao Chen

Final

1. For each of the following answer T if the statement is always true and F otherwise. In each

case A, B ∈ Rn×n .

(a) If A and B are row equivalent then det(A) = det(B)

(b) If Nul(A) = {~0} then Col(A) = Rn .

1

(c) If n = 3 and A 3 = ~0 then rank(A) ≤ 2.

2

(d) det(AB) = det(A) det(B)

(e) If B is created by swapping two columns of A the det(A) = − det(B)

(f) If A and B are row equivalent then Col(A) = Col(B)

(g) If Col(A) = .Rn then Nul(A) = {~0}.

(h) If A has an eigenbasis for Rn then A is invertible.

2. Let:

−4

3

0

A=

0

0

6

0 0 0 0

1 0 0 0

0 −3 0 1

0 −2 1 0

0 −2 0 0

2 0 0 0

−12

7

−2

~b =

1

−4

14

(a) Find all solutions to A~x = ~b.

(b) Are the columns of A linearly independent?

(c) Do the columns of A span R6 ?

3. Consider:

−3 6

3 0 0

3

2 −4 −2 1 0 −1

6 −12 −6 3 −2 7

A=

−3

6

3

0

−1

8

6 −12 −6 0 −2 4

8 −16 −8 1 0 −7

which it row equivalent to:

1 −2 −1 0 0 −1

0 0

0 1 0 1

0 0

0

0

1

−5

rref(A) =

0 0

0 0 0 0

0 0

0 0 0 0

0 0

0 0 0 0

(a) Find a basis for Col(A).

(b) Find a basis for Nul(A).

(1)

(c) Find rank(A)

(d) Find nullity(A)

(e) Are the columns of A linearly independent?

(f) Do the columns of A span R6 ?

4. Let T : P3 → R2×2 be defined by:

a + 2b c

T (ax + bx + cx + d) =

3d

2c

3

2

You may assume T is a linear transformation without proof.

(a) Find a basis for ker(T ).

(b) Find dim(ker(T )).

(c) Find dim(Rg(T ))

(d) Find a basis for Rg(T )

(e) Is T onto?

5. Let:

−3 −4

A=

0 −1

.

(a) Find all the eigenvalues for A.

(b) For each eigenvalue, find a corresponding eigenvector.

(c) Is A diagonalizable?

6. Let:

and

2

0

0

A = 71 −17 30

38 −10 18

1

0

0

B = −3 , −1 , −2 .

−2

−3

−1

(a) Show that B is an eigenbasis with respect to A and find the corresponding eignenvalues?

Make sure to explain how you know it is a basis.

(b) Find D and P so that AP = P D.

7. Now that you have finished, scan this with an app like Genius Scan or even an actual scanner

to turn this into a pdf file (not a jpeg). Name the file “final.pdf”. Put this file in our shared

directory (not the test subdirectory). Do all this to get 2 points on the exam.

Page 2

Purchase answer to see full

attachment

Tags:

Math Problems

Matrices

math worksheet

Row equivalent

Nullity Theorem

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.