MATH 18 University of California Berkeley Linear Algebra Practice Questions

The main source of course material is the textbook, Linear Algebra and Its Applications; Fifth Edition (Lay, Lay, & McDonald).link:

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

cover Sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8, 2.1 Linear
APgebra
Linear equation
32
non
32449
7
44
linear equation
sense
3K
KATY
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coefficients
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t a
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ant
a
ay
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solution
Linear equation
System of linear equations linear
system
Solution set
coefficients
equivalent system are systems
that have the same solution set
22
ya
47
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129
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se
222
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se
322
3
42
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No
22
102,41
22,43
solution
I
I
3
d Ilk
solution
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
solution
many solution
infinitely
3
8
2
222
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y
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a
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ans
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off
3
4
the system has
a
matrix
entries
a
row S
n
columns
of
a
matrix
en
en
e
a
a
a
coefficient matrix
Augmented matrix
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
found that
unique solution
solo
1
11
a
has
a
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
0
I
2
4
88 9
24
8
1
f
kitted
e
OH
the system
we
have
solution
a leading
number in
is
022 023
INCONSISTENT
been
applying
process
efyÉ
a
row
that
is
t
a
called
non
zero
15
1
is
chelon form
form
the
matrix
A
Definition
ray e halon
it has
in
if
following properties
All the
any
non
zero
rows
are
above
K
zerwae
2
Each
entry of
leading
a column
it
3
All
the
below
in
leading
zeros
2
entry
entries
a
column
entry are
a
1
t
I
row
to the right of W
of the row
is in
the Pending
above
a
1
echelon form
Me
c
G
Reduced rechelon form Crreff
a
ref form if
A matrix
it
is in echelon form and
4
the leading entries
5
All the entries in
leading
a
are
are
a
all
column
zeros
I
I
To
0
3
0
I
0
0
08020
0
0
0
o
o
ref
food
o
o
o
d
5 6
i
s
above
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
ÉÉÉ
Either
column
p
ending entry
entry in a matrix that
at the same position
corresponds
pivot
to
is an
leading entry
a
in
an
chelon
form of the matrix
A
a
X
column
with a pivot is
pivotcolumn
Xz
Xz Xu
as
called
H
Az
I
Trm
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2 ox
q
1
I
0
o
o
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r
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t 8Xy
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in
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leading
r
e
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IF
sees
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see
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a
a
se
se
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t
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3
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t
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18
30
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Ir
tax
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3
g
1ÉFFEY
P
r
t
t
4
1
11
11
Eam
parametric form
infinitely manysolution
vector form
Parametric form
o
basis
free
variable
variable
o
of
theorem
A
at
not
A
linear system
a
row
o
I
the
o
consistent off
system does
of the form
echelon form of
have
a
the
nightmost column of
D
to
o
the augmented
Eto
Linear
system
Inconsistent
Consister
a
free variables
infinitely
many
No
free
variable
d
unique
solution
solutions
theorem
A
linear system either
has
solution infinitely many
solution or any one solution
no
Sector l I
a
Every elementary operation
239
invertible
ÉE
E
EÉ
t.it
LIFE
At
29
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2b
1C
2ETE
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p
at
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btc
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10
Construct three
26
different
for
augmented matrices
linear system whos solution set
2
2
32
1
33
285
31
O
f
Mr
U
0
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we
shouted
only solution
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X
the
2
42
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2
is
22
21
322
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43
2
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322 f
da
g
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ca
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can
solution
2
en
3
t
consistent for all
values of f and g
you say about
e
and d
dreg
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at
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l
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8
that
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Hence
can
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say
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that
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Solve
o
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l
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s
it
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3
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free variable
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023
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5
21
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r
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t
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tr
j
5
Gt
tree variant
5
Gt
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IF
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infinitely many solutions
It
23
Suppose
345
a
as
coefficient matrix
system has three pivot columns
Is the system consistent
for
a
FINI
ÉE
or
of thee
rows of an
chelor form
none
of theft’atrix
o o
o
o
ul
is
a
zero raw
to
TINA
pivot column is
to a column
a
column that corresponds
in an chelon form with
a
leading entry
Leading entry a the first non zero entry
in
a
row of an Cholon form of a matrix
two leading entries exist in one
no
cola
column
Therefore each
matrix
has
Hence
we
of the form
row
efficient
of the
pivot position
a
do not
o
o
get
o
a
o
row
o
to
So
the system is consistent
133
71
t
space
g
y
i
is
R2
g L
of
the set of
Tall
column vectors
the possible
pi CR
called
the
set of
3
two
all possible column
f
is
vectors
3
pi
is
the
R three
set of all possible
form
of the
i
call
column vectors
RAI
pi
n 1
a
n
3
line through zero
plan through zero
Space
1 41 11
af ft
w
Efarmullirlicate
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Melo
Law
Parallelogram
a
11
went
o
v
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au
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In
Algebraic properties of R
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c
linear
system
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f
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tht
bun
am
a
unknowns
n
equation
92
b
t
entry
ith
and the
column
1 I
Tierra
as
in
row
jth
q
i
Am
Amr
T
T
T
Ambatrix equation
define
the multiplication of an
A by a column rectory
mxn
matrix
in R
to be
we
ax
xia.EE
inIIafa
columns
say that the system has a
solution is the same as
saying that
the vector equation has a solution
To
b must be
columns
as
in
relation
e
with the
in
the
a
expressed
is
called linear combination
vector equation
this relation
theorem
A
system
A lb
has
b is a linear
combination of the columns of A
a
solution
if f
Linear
Construct
was
3
a
linear system
vector
the
LIE
in
4
combination
which
É
EH HE
w
er
011 011 01
we
or
I
a
É
b
is
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a
linear combination of
Az Az
É
T
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eights of the
linear combination
Let
ve
Question
f
b
Is
of
Solution
linear
a
combination
V2
u
there
are
f
b
L
v
V
K
T
hunters
X
x
b
Yak
E 47 1
X T 2
Z
2
24
5 2
54
to Xz
4
3
It
1
Yes
Thratia
7
b
is
a
É
1
d
c
Xz
2
X
3
of
y
re
but
Exod
Is
E
A
f
consisted
btba.bz
the equation
b
Ax
all posiate
for
E I
b
Solution
I
t
1
the system
b
is
example
then
the
for any
No
iff
bet 451
0
by
that
2b 432
For
consistent
35
b
2
0g
053 33 213144
253
0
62
b
147
0
42
7
53
0
040
system is not consistent
choice of
b
baby
I
v11
spans
G
I
Lev
d
a
c
of
i
v
All the vectors
line
the
y
X
all the possible linear continalis
are
of
on
E
a
v
L
s
a
s
Ey
spans the
a
se
axis
i
EE CE
I
a e
take
29
39
the
set of
all d
80
9 ez
8
E
ae
t
of
c
3
319
243
e
ez is pi
be
U
avi
Y
u
t
a v
o
t b ok
11
K
S pan
R2
11
weight of the he
rest
92,9
Spangs
1
Kit’d
9
Q
beetle
sac
Zn
g
redo
v
su
n
not span
M3
G
E
u
a
nota d
Vi V2
Id’d
me
o
c
of
b is not
Definition
the set of
risk
up
Let
u
doc
a
spanned by
Vi Vas sup
a
Vio Uz
Vptvector
ve
all linear
in
of
in
combinations
subset of
R
of
that
R
Vi Vas
Ea
sup
this subset
generate
Each vector
of
Va Va
v
in the
set is
a
d
c
Up
A V
92k
tu
tapUp
V1of the combination
weight
is
4
3
7
31
I
b is
4
a
7
Eg
Yu
doc
are
of
V
vector
T
and ve
the weights
of the
t
A
1
x
A1
form
ax
FIVE. o
I
combination
I
X
5
2
i
343
Theorems
Then the
Let A be an mxn
matrix
e
following statements are logically
equivalent
a
b
For each be Ri A x b has
Each b ERM is a doc of the
columns
c
d
of
a
solution
A
The columns of A span RM
A has
a
pivot position
in
every row
Aman
and
v
Ih
w
are
1
2
solution to the system
b
Av
Lin
berm
awes
Is
vew
Is
ru
a
solution
solution
a
d
scalar
T
b
T
O
But if 3
Ex
3
ru
Tawes
b
2b
If
A
b
Ib
Acuta
a
A Crew
then the
0
answer
then the
is to
answer
is
yes
b
Ase
th
Ésoafd
system
I
Non homogeneous
system
with
not be
May
consistent
Always consistent
DEO
Ao
a
fax
iff b
a consistent
b
O
is
of the
a
columns
5
1
A
d
c
ofA
1
b
Ax
inconsistent
É
t
a
b
Ax
consistent
free variables
No free variables
d
unique
d
infinitely
solution
Question
solutions
Ax
b
Ax O
solution sets
what
is
S
many
Consistent
Ax
b
solution set T
the relationship between
and
T
Example
15
s
4
A
X
AX
O
2
x2 He
22
32
32
32
4
2
0
0
23
U
423 824
522
5 2 t
10
E
b
y r
St
to
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e
7
7
7
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2
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It
Yr
430
4
1
1
V
14
1
I
i
solutions of
Take
t
o
r
0
one
vector
it
4
Ed
f
HEFEI
EEFEHE.EE
viaxsoluEonFAaI
I
Particular
are
Y re
solutions
Ase
aft
particular
solution to
Ax
b
Axe
O
b
t
Etty
all thesolutions
t
An
0
red
É
ttivenittik
Iv
ta
all
th
where
scalars
be
flee
a
non
Tom
general solution to
System
b
Ax
Thank
is
particular solution
a
to Ase b
set
and
where ti ti
ttzvetnitt.dk
th are scalars
V
generalsolution
is the
hom system
Iv
V4
D
Tent
ta
Elk
A
has only
x
Ax
0
as
b
the
0
Ax
b
Ax
t
a
has
solution
a
unique
solution
Question
Under
hom
zero
what conditions
a
Dy the
system has
on
solution
ay
I T T
3 A 44 Az
s
0
x
no
EMEI
III
O
F’solution to
cola
Avo
is
Since the third column A
a doc
of the first and second
columns A Az the AX 0
has a
Therefore
non
zero
has infinitely
it
solutions
solution
many
0
4,3
Etitt
I
Say
a
non
we
know
that
solution
zero
E
Axe
to
meaning
Lai
I
has
one
the entries is
qq.to
of
non
zero
IA
Iz Az
IGA
Az Az
9 A
G A
Iu Au
O
Ay Ay
T
es
Az
Atf
Eas
e
Ax
o
has
a
then
solution
columns of A
is
non
a
d
Axe
zero solution iff
columns
other
of A
columns
is
a
of A
has
one
zero
of the
one
the rest
Conclusion
G Ay
Azt
c
of
a
non
of the
doc of the
Axe has only the
getrolution eff none of its
columns is not a d c of the rest
Conclusion
Definition
Let
that
Tesay
dependent if
Vi Vas
us
ra
one
Vk
he
are
of them
linearly
is
a
linear combination of the rest
are
We
that in vz
ye
linearly
say of none
is
vectors
the
independent
of
a
linear combination of the rest
µ
ref
t
V2
I
III I
4
44
are
it
4
k
Examf Determine
483
1
are
d
of the vectors
L
i
solution
We want to
is a doc
we Pearned
is a doc
know
of the
that
of
of the vectors
one
rest
one
of the vectors
of the rest iff the
system
a
1
non
Y
I
E
x
has
0
zerosolutionTA
LIFE
I
1H I
1
1
É’d
No
has
free variables
only
the
columns
of the
of the rest
none
Hence
zero solution
of as
a
unhryaillac
doc
Example
IIT IllIT
are
er
Ty
us Va Va Va
doo
I
L
Must
Hence
have
Ax
a
0
many solutions
Therefore
we
feast one f u
has infinitely
re us by
al
lo d
Theorem
If
p
Let
R
us vz
then
u
up
V2
Vp
live.it
P n
Hence there
vector
the matrix
eseistfau.mn
is
are
short
in
R
dad
Matrix Algebra
Bryn
Amen
I
A
a 43
A
I
1
13
8 38
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At
B
IBM
43 1
1
433 828
L’s
3
2 2
3
273
Thighs B
same
be matrices
and let r and
c
size
scalars
a
At B
B
b
At B
C
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c
A
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At
A
Zero matrix
de
B
r
e
rts
f
re A
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ra
ra
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t
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of
s
the
se
Multiplication of
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w
tat
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o
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AB
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matrices
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n
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AB
AB
43 3
11
273
2 2
1
AB
D
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column multiplication
Row
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443
I
342
i
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3.7
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g
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u
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16
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87
T
416
E 26
38
1.1
1
E
1
6
u
E
u
Let
Teared
A
be
an
and let B and
for which the following
matrix
product
I
defined
are
ACB C
CAB
A CBT C
23
3
CBTC
4
r
A
AB
CAB
In A
In
TAC
B A TCA
B
scalar
A
r as a
5
C
A
AIn
18
identity matrix
man
matrices
sum
and
C
A
I
E
L
IA
9
38
38
AI
383
9
38
Warnings
1
In
2
AB
3
AB
general
AC
o
BA
AB
A
B
C
0
or
Bo
D
EEE
FEI
A
At
Transpose
A1
Symmetric
matrices
ATBT
CAB
CABS
B’AT
be
Theorems Let A and B
are appropriate
matrices whose sizes
for the following sums and products
AT
A
CATBF
AT TBT
GAF
r
CAB
r
es
AT
a
scalar
B’AT
Theorem
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it is
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of
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matrix
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unique
Theorem
Let A
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matrix
T
CATIA
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f
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pay
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pay
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matrices
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and
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o
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let A
theorem
matrix
matrix
square my
The following statements are
be
n
a
logically equivalent
a
A is
b
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C
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d
e
invertible
has
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equivalent to
row
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o
pivot positions
has only the zero
n
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g the equation Ax
b
h
J
K
l
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d
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has
at
least one solution for each b ER
the columns of A span R
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AT
a
matrix C
n xn
CALI
nyn
matrix D
AD
invertible
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59
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has only
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disc
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only
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the
has
pivot position
n
solution
proof
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it has
0
no
has only the zero Solution
free variable
Therefore each column of
a
pivot column
Since A
positions
is
A is
it has
nxn
n
pivot
cab
A chas
A
n
pivot positions
BEGA
to
ataivalat
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rreffa has n leading Ms
rref CA
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b
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to
row equivalent
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invertible
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the algorithm explained above
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b
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a
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AX
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s
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has a solution
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inv
for each beR
Proofartiparly
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call it Bi
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u
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call it Bz
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en
e
call it Bn
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en
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invertible
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I
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A x b has
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the columns of A
span R
each be pi
has
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a
solution
for each b
in
es
for each b a R’s
b is a doc of the
columns of A
Rr
each b er is in span of
the columns of A
the
columns
of A span R
dese
the columns of
the system Axe
has only a
solution
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we
prove
A
this
on
are
d i
Monday
tea
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a
proof
invertible
see
Ai
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above
ex
invertible
Page 1 of 6
Print Name__________________________PID:__________________________
Last
First
Math 18, Midterm Exam, Summer 2021
Duration: 2:30-4:00
Show all your work; no credit will be given for unsupported answers based on the material
presented in the lectures.
1. What relation must 𝑎, 𝑏, and 𝑐 satisfy so that the following system is consistent for all values
of ℎ and 𝑘?
𝑥 𝑎𝑦 ℎ
𝑏𝑥 𝑐𝑦 𝑘
Page 2 of 6
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2. Let 𝐴 be an 𝑚 𝑛 matrix and let 𝑏 be a vector in 𝑅 .
a. If the system 𝐴𝑥 𝑏 has a unique solution, must the system 𝐴𝑥
b. If the system 𝐴𝑥 0 has a unique solution, must the system 𝐴𝑥
0 have a unique solution?
𝑏 be consistent?
Page 3 of 6
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3. Let 𝑀 be an 𝑚 𝑛 matrix. Supposed 𝑀𝑣
𝑢 and 𝑀𝑣
𝑀𝑥 𝑟𝑢
𝑡𝑢 consistent for all scalars 𝑟 and 𝑡?
𝑢 . Is the system
Page 4 of 6
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4. Let 𝐴 be a 7 9 augmented matrix of a system of linear equations and let 𝐵 be the coefficient
matrix of the system.
a. If the 9 column of 𝐴 is a pivot column, is the system consistent?
b. If the 7th row of an echelon form of 𝐵 has a leading entry, is the system consistent?
Page 5 of 6
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5. Let 𝑣 , 𝑣 , 𝑣 , 𝑣 be vectors in 𝑅 , where 𝑣 , 𝑣 , 𝑣 are linearly independent and 𝑣 is not in the
𝑆𝑝𝑎𝑛 𝑣 , 𝑣 , 𝑣 . Must 𝑣 , 𝑣 , 𝑣 , 𝑣 be linearly independent?
Page 6 of 6
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6. Construct a non-homogeneous system with 3 equations and 3 unknowns, such that all the
1
entries of its coefficient matrix are different from each other and 𝑥
2 is a solution of the
3
system.