EE103 Lecture Notes, Fall 2007, Prof S. Jacobsen
Section 4
47
clear that the relatively small error in the value of
2
x
(1.001 instead of 1.000) is being
greatly inflated because of the division by the number .0003.
Ex:
Consider the system of linear equations
11 1
12
2
1
1
11
nn
n
n
n
ax ax
ax b
ax
+
++
=
+
+=
...
.................................
..............
or, in matrix-vector notation,
A
xb
=
Basic Linear Algebra Facts
You've all learned that if
A
has an inverse, denoted by
1
A
−
, then the unique solution is
given by
1
x
Ab
−
=
*
.
However, the latter is not meant to be a prescription for a numerical
method for computing the solution
x
*
.
That is, we'll see that there are better methods,
especially for large systems (many equations and variables) and, in particular, one should
not generally try to find
1
A
−
and then execute the multiplication
1
−
.
In fact, suppose
we have another method for solving the
x
system of equations
Ax
b
=
.
We can use
that method for computing the inverse matrix.
To see this, let
12
n
ee
e
,
,...,
denote the unit
vectors in n-space.
That is,
0 0
0 1 0
0
i
e
=
( , ,.
.., , , ,.
.., )'
, where the "1" is in the i
th
position.
Then, we use the method to solve the n linear systems of equations
1
i
Ax
e i
n
==
,
Let
1
i
x
in
=
,
denote the respective unique solutions.
Then the matrix
n
x
xx
[ ,
]
is the inverse of
A
.
To see this, note
1
1
n
nxn
Ax x
x
Ax
e
e
I
=
[ ,
] [
] [ ,.
..,
]
Much of our subsequent discussion will involve the development of methods of solving
systems of linear equations.
However, those methods are based upon some important
concepts of numerical linear algebra.