1
15.053
February 7, 2002
z
A brief review of Linear Algebra
z
Linear Programming Models
Handouts:
Lecture Notes
2
Review of Linear Algebra
z
Some elementary facts about vectors and
matrices.
z
The GaussJordan method for solving
systems of equations.
z
Bases and basic solutions and pivoting.
3
Elementary Facts about Vectors
[
]
1234
vvvvv
=
is called a
row vector
.
1
2
3
4
=
t
v
v
v
v
v
The
transpose
of
v
is a
column
vector
.
[
]
wwwww
=
is another row vector.
The
inner
product
of vectors
w
and
v
is
given by:
11
22
33
44
v w vw vw
vw
=+
++
D
4
Matrix Multiplication
()
=
ij
A
a
=
ij
Bb
==
×
ij
Cc AB
1
=
∑
=
k
kj
n
ik
j
i
ab
c
Suppose that A has n columns and B has n rows.
5
Multiplying Matrices
Let C = (
c
ij
) = AB.
Then
c
ij
is the
inner
product
of row
i
of
A
and column
j
of
B
.
21
22
23
11
12
13
31
32
33
aaa
A
=
13
23
11
12
21
22
31
33
32
b
b
b
bb
Bbb
=
For example,
what is
c
23
?
23
21 13
22
23
23 33
ca
b
a
ba
b
=++
6
Multiplying Matrices
Let C = (
c
ij
) = A
×
B.
Then each column of
C is obtained by adding multiples of
columns of A.
1
2
3
100
123
456
1
0
4
5
6
7
8
9
1
789
×=
12
3
100 4
10 5
6
78
9
+
Similarly, each
row of C is
obtained by
adding multiples
of rows of B.
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View Full DocumentElementary Facts about Solving
Equations
12
3
4
0
21
16
xx
x
++
=
−
Find a linear
combination of
the columns of
A
that equals
b
.
Solve for
Ax = b
, where
2
×
1
12 4
21 1
A
=
−
1
2
3
x
x
x
x
=
0
6
b
=
2
×
3
3
×
1
123
240
26
xxx
++=
+−=
8
Solving a System of Equations
To solve a system of equations, use GaussJordan
elimination.
x1
x2
x3
x4
1241
=
0
21
1

1
=
6
1
1
2
2
=
3
9
The system of equations
1
2
1
2
1
1
4
1
2
1
1
2
=
=
=
0
6
3
x
1
x
2
x
3
x
4
10
Pivot on the element in row 1 column 1
=
=
=
0
0
3
3
9
6
3
3
1
2
4
1
0
6
3
Subtract 2 times constraint 1 from constraint 2.
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 Spring '05
 Prof.JamesOrlin
 Linear Algebra, Dot Product, x3, 1, Howard Staunton, Postal Workers

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