Some Linear Algebra Notes
An
m
x
n
linear system
is a system of
m
linear equations in
n
unknowns
x
i
,
i
= 1
, . . . , n
:
a
11
x
1
+
a
12
x
2
+
· · ·
+
a
1
n
x
n
=
b
1
a
21
x
1
+
a
22
x
2
+
· · ·
+
a
2
n
x
n
=
b
2
.
.
.
=
.
.
.
a
m
1
x
1
+
a
m
2
x
2
+
· · ·
+
a
mn
x
n
=
b
m
The
coefficients
a
ij
give rise to the rectangular matrix
A
= (
a
ij
)
mxn
(the first subscript is the row, the
second is the column). This is a matrix with
m
rows and
n
columns:
A
=
a
11
a
12
· · ·
a
1
n
a
21
a
22
· · ·
a
2
n
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
.
A
solution
to the linear system is a sequence of numbers
s
1
, s
2
,
· · ·
, s
n
, which has the property that
each equation is satisfied when
x
1
=
s
1
, x
2
=
s
2
,
· · ·
, x
n
=
s
n
.
If the linear system has a nonzero solution it is
consistent
, otherwise it is
inconsistent
.
If the right hand side of the linear system constant 0, then it is called a
homogeneous
linear sys
tem. The homogeneous linear system always has the
trivial solution
x
= 0.
Two linear systems are
equivalent
, if they both have exactly the same solutions.
Def 1.1:
An
m
x
n
matrix
A
is a rectangular array of
mn
real or complex numbers arranged in
m
(horizontal) rows and
n
(vertical) columns.
Def 1.2: Two
m
x
n
matrices
A
= (
a
ij
) and
B
= (
b
ij
) are
equal
, if they agree entry by entry.
Def 1.3: The
m
x
n
matrices
A
and
B
are added entry by entry.
Def 1.4:
If
A
= (
a
ij
) and
r
is a real number, then the
scalar
multiple of
r
and
A
is the matrix
rA
= (
ra
ij
).
If
A
1
, A
2
, ..., A
k
are
m
x
n
matrices and
c
1
, c
2
, ..., c
k
are real numbers, then an expression of the form
c
1
A
1
+
c
2
A
2
+
· · ·
+
c
k
A
k
is a
linear combination
of the
A
’s with coefficients
c
1
, c
2
, ..., c
k
.
Def 1.5: The
transpose
of the
m
x
n
matrix
A
= (
a
ij
) is the
nxm
matrix
A
T
= (
a
ji
).
Def 1.6: The
dot product
or
inner product
of the
n

vectors
a
= (
a
i
) and
b
= (
b
i
) is
a
·
b
=
a
1
b
1
+
a
2
b
2
+
...
+
a
n
b
n
1
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Example: Determine the values of x and y so that
v
·
w
= 0 and
v
·
u
= 0,
where
v
=
x
1
y
, w
=
2

2
1
,
and
u
=
1
8
2
.
Def 1.7:
If
A
= (
a
ij
) is an
m
x
p
matrix and
B
= (
b
ij
) a
p
x
n
matrix they can be multiplied
and the
ij
entry of the
m
x
n
matrix
C
=
AB
:
c
ij
= (
a
i
*
)
T
·
(
b
*
j
)
. Example: Let
A
=
1
2
3
2
1
4
2

1
5
and
B
=

1
2
0
4
3
5
.
If possible, find
AB
,
BA
,
A
2
,
B
2
.
Which matrix rows/columns do you have to multiply in order to get the 3
,
1 entry of the matrix
AB
?
Describe the first row of
AB
as the product of rows/columns of
A
and
B
.
The linear system (see beginning) can thus be written in matrix form
Ax
=
b
.
Write it out in detail.
A is called the
coefficient matrix
of the linear system and the matrix
a
11
a
12
· · ·
a
1
n
.
.
.
b
1
a
21
a
22
· · ·
a
2
n
.
.
.
b
2
.
.
.
.
.
.
.
.
.
a
m
1
a
m
2
· · ·
a
mn
.
.
.
b
n
.
is called the
augmented matrix
of the linear system.
Note:
Ax
=
b
is consistent if and only if
b
can be expressed as a linear combination of the columns of
A
with coefficients
x
i
.
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 Spring '08
 Neely
 Linear Algebra, Vector Space

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