# 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...

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