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Math 1104C, 1104J: Assignment # 3, Due April. 6 11:59, 2022

Directives:

• This document has 2 pages (including this page).

• This assignment is due on April 6th at 11:59 p.m. on Brightspace.

• The assignment has 8 questions for a total of 57 points.

• You must show your work when appropriate. We look at lot more at your work than you

final answer and we need to see your work when asked.

IMPORTANT:

Submit a scan or pictures of your work. You can write all your answers on a piece of paper of your choice

or an electronic tablet or use latex.

Instructions to submit your assignment online

• You must send a scan or a picture of your work. It must clear enough so that we can read what you

wrote. We are only marking what we receive so make sure to send complete files.

• only .pdf and .jpg files are accepted.

• Make sure to submit your work at the appropriate link of the exam on Brightspace.

• Only the last version of a file or document that you submit is marked.

• Please submit your work as only one file or folde

• No late work will be accepted.

You can contact me by email at XXXXXXXXXX.

1. (4 points) Find two independent vectors both of which are perpendicular to ~v =

12

3

.

2. Let

A =

(

5 −6

3 −4

)

.

(a) (6 points) Find the eigenvalues and eigenvectors of A.

(b) (2 points) Diagonalize A.

(c) (4 points) Find a formula for the entries of An.

3. (6 points) Find the eigenvalues and co

esponding eigenvectors of A =

[

−1 1

−1 0

]

.

1

mailto: XXXXXXXXXX

4. (10 points) The eigenvalues and eigenvectors for the matrix M are given, in no particular order. You

task is to diagonalize the matrix M , and use this to compute M5.

M =

[

4 −2

1 1

]

. Eigenvalues: 2, 3 Eigenvectors:

(

2

1

)(

1

1

)

5. (6 points) Consider A =

− XXXXXXXXXX

4 −1 −1

.

(a) Find all eigenvalues of A.

(b) One of the eigenvalues is λ = 2. Find the co

esponding eigenspace for λ = 2.

6. (4 points) Consider the complex numbers:

z = 2− i and w = 1 + 3i .

Sketch (in the same diagram)

z, w, zw and z̄ ;

7. (6 points) Let w = 1 +

√

3i.

(a) Sketch w on the complex plain.

(b) compute w7.

8. (a) (3 points) Find the polar form reiθ for i.

(b) (6 points) Find the three complex cube roots of i. (This means solve u3 = i in polar form.)

Page 2

Answered Same DayApr 06, 2022

SOLUTION.PDF## Answer To This Question Is Available To Download

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