Linear Algebra and Eigenvalues Questionnaire

Book:

https://math.berkeley.edu/~yonah/files/Linear%20Al…

Section 1.1: Problems 1-5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25-27.

Section 1.2: Problems 1-3, 9, 11, 13, 17, 19, 23-26, 28-31.

Prove Theorem 5 on Page39

Section 1.3: Problems 1, 2, 5, 6, 11, 12, 13, 14, 15, 17, 21, 23, 25.

Section 1.4: Problems 1-6, 9-12, 14-22, 25-26, 29-36.

Section 1.5: Problems 1-5, 7, 9, 11-19, 23-31.

Section 1.7: Problems 1-3, 7, 9, 11, 17, 21, 23-29, 31-40.

Section 2.1: problems 1-12, 15-26 (only the odd-numbered problems)

Section2.2: Problems 1-5, 7, 8, 11, 13, 15, 17, 18, 29-33.

Section 2.3: Problems 1, 3, 5, 7, 15, 17, 19, 21-24, 26-28.

Section 2.8: Problems 5, 7, 8, 11, 13, 15, 19, 20, 23-25, 27-36

Section 2.9: Problems 1, 3, 5, 9, 19-26

Section 3.1: Problems 1, 3, 5, 7, 9, 11.

Section 3.2: problems 1, 3, 5, 7, 9, 15, 17, 19, 21, 23, 25, 28, 31, 33, 35, 37, 39, 41, 43

Section 5.1: Problems 1, 3, 5, 8, 13, 15, 23-27,

Section 5.3: Problems 1, 2, 5, 6, 7, 15, 21, 23-32

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coefficients
equivalent system are systems
that have the same solution set
22
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129
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se
222
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102,41
22,43
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I
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3
d Ilk
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No
the solution set is empty
a
21
222
1
222
10
infinitely
of
many
solutions
observation
Geometric
system of linear equations
A
1
has
2
exactly
no
one
22
f
0
8
823
s E 4
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infinitely
3
8
2
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3
4
the system has
a
matrix
entries
a
row S
n
columns
of
a
matrix
en
en
e
a
a
a
coefficient matrix
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size of
a
matrix
example
J
cofficient matrix is 32 3
Augmented matrix
d
by
92
get
are theft Cat dz
93kt by C Z d
Replacement Replace
II
represents
a
plane
ee
en
Operations applied
size
Elementary operation
one
row
by
the
of itself and a
multiple of another row
sum
interchaning two
Interchange
scaling
multiplying
a
THESE
DO
number
ELEMENTRY
NOT
CHANGE
one
row
rows
by
different from zero
OPERATIONS
THE SOLUTION
SET
the operation keep the systems
equivalent
A system
that has
called
consistent
A
a
a
solution
system
system that does not
solution
is
is
called
have
inconsistent system
Questions
Given
a
Is
If
have
linear system
the system consistent
yes
a
does the system
unique
solution
Exampf Determine of the following
system
is
consistent
222
23
0
222
823
8
2
see
5
10
3
found that this system es
We
consistent
es
AP so
a
23
I
22
0
se
A
we
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solo
1
11
a
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Determine
in
consistent
of the
following
system
423 8
32 223 1
22
22
822 1223 1
42
I
I
2
3
2
2
3
ii I
I
1
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2
4
88 9
24
8
1
f
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we
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a leading
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is
022 023
INCONSISTENT
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efyÉ
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t
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15
1
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chelon form
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the
matrix
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Definition
ray e halon
it has
in
if
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any
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above
K
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2
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it
3
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the
below
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column
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a
1
t
I
row
to the right of W
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the Pending
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Me
c
G
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a
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it
is in echelon form and
4
the leading entries
5
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a
all
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o
o
ref
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o
o
o
d
5 6
i
s
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Theorem
determine if a system is consistent
reduce it If at any stage
get
chelon form where a row looks like
To
row
an
o
o
to
o
thesystemisinconsistent
leading entry
then
I
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Either
column
p
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at the same position
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pivot
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a
in
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chelon
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A
a
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column
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e
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t
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at
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o
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the
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D
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o
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Linear
system
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Consister
a
free variables
infinitely
many
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free
variable
d
unique
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solutions
theorem
A
linear system either
has
solution infinitely many
solution or any one solution
no
Determinants
d
a set
a
da t
d
by
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b
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db
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tae
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azz
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e
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I
ax
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We can compute the determinant of a
squarematria cross any row and along
any column
EID
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j
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E
sit I
41
4
0
T
o
I
I
L
9
1
1
1
2
L
ii
tatted
9
far Ian
9
92
an
theorem the determinant of a
diagonal matrix is the product
of the entries on the main diagonal
of the matrix
A
L
data
1 3 C
5
15
Theorem
the
determinant of
upper triangular
or
an
lower
matrix as the product
of the entries on the main diagonal
triangular
FEET
8
deta
7 t.lu
det 13
73 18.14
L
1
U
Z
H
date
detB deta
d etc
1
0
I
03
3
upper
I
tÉ
det B
D
L
0.2
2
triangular
6 57.3
15
Theorend Let
1
If
B
is
adding
a
A
be
a
obtained from A
multiple of
then
another row
matrix
square
row
by
column
to
detBideta
2
B is obtained from A by
rows
then
interchanging two columns
If
data
detB
3
If
B
is
multiplying
hunter
r
obtained from A
row
by
thalamus
one
detBerdeta
a
by
non
zero
Ann
upper triangular
Edette
deta
data
tefft
dette
to
O
Unn
Un Un
is
A
TITTA be
A
a
and
O
invertible
square matrix
invertible
TITTA
Then
a
Es
B
detCAB
iff
be
nen
data
detato
matrices
dB
Review
we
we
define
dot
this
use
define dot
We
stated
between
of
2
matrix
2
definition
of
men
to
matrix
the relationship
the dot of
a
matrix and the determinant
of a matrix that obtained
from it by elementary operations
on
row
columns
compute det
any column and across
We
Al
can
0
off
dat CAB
A
along
any row
invertible
data
etB
problem
the redwood forest of California
spotted owl is the main predator of the
wood rate
In
month k
given
Say
the population of the owls
en
Ii
en
rats
w
a
e
Ia
la
Ta
Lk
5
t a 4
Go 10474
as
l
as
Tk
T
C 17Th
Find the evolution of the population
of the owls and rats is the redwood forest
E T in
x
y
Kel
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f
IED
Name: __________________
Last
First
UCSD ID #:____________________
Math 18 Summer 2021
2 hours
1. No materials and calculators are allowed during the exam.
2. Write your solutions clearly and (a) Indicate the number and letter of each question; (b) Present your answers in the same order they appear in
the exam; (c) Start each question on a new page.
3. Show all your work; no credit will be given for unsupported answers.
4. Your answers must be based on the material presented in class; no credit will be given otherwise.
5. We reserve the right to examine you in person on your answers to the problems in this exam.
1 0 1 


1
1. Let A  0 1 0 Show that A is diagonalizable, and write it as A  PP . Find P and the diagonal matrix  . You do


1 0 1 
1
not need to find P . (15 Points)
2.
a. Construct a
3  3 matrix that is invertible but not diagonalizable. (10 Points)
b. Construct a non-diagonal
3  3 matrix that is diagonalizable but not invertible. (5 Points)
3. Let A be diagonalizable. Prove that the determinant of A is equal to the product of its eigenvalues. (10 Points)
4.
a. Construct a
Points)
3  3 matrix A with no zero entry and a vector b  0 in R 3 such that b is not in the column space of A . (10
1
 
b. Construct a 3  3 matrix A with no zero entry such that the vector b  1 is in NulA . (10 Points)
 
1
 4 5 9 2 


5. Let A  6 5 1 12 . An echelon form of A is the matrix


 3 4 8 3
a. Find a basis for
1 2 6 5
0 1 5 6  .


0 0 0 0 
ColA (5 Points)
b. Find a basis for NulA (10 Points)
6.
3
 1 2 3




a. Find the orthogonal projection of b  1 onto the column space of A  1 4 3 (15 Points)
 


 5 
 1 2 3
b. Find all the least square solutions of
Ax  b . (10 Points)
Problems 1-8 (Note u-v=ufv = vt u)