Read for Friday. Section 1.7 on inverse matri-
ces.
Due Friday. Exercises from section 1.5: 1-2, 5-
6, 15-17.
Due Monday. Exercises from section 1.6: 1-8,
13-14, 20, 28, T8.
Last time. Linear transformations , also called
matrix transformations.
Today. Solutions of linear systems of equations .
We discussed this before on the first day, and you're
already familiar with it from high school courses,
so we won't spend too much time on it. Know
what "echelon form" means, what " reduced echelon
form" means, and how to solve systems of linear
equations. We'll look at an example of a system
where the solution is not unique so you can see
how you can describe the set of its solutions.
We may also look at the preview of graph theory
on page 48.
Elementary row operations, and row echelon
and reduced row echelon form for a matrix.
When you want to solve a system of linear equa-
tions Ax = b, form the augmented matrix by ap-
pending the column b to the right of the coefficient
matrix A. Then you can solve the system of equa-
tions by operating on the rows of the augmented
matrix rather than on the actual equations in the
system . The three row operations are those oper-
ations on a matrix that don't change the solution
set of the corresponding system of linear equations.
The simplest ones are called elementary row oper-
ations, and the elementary ones are of three types.
1. Exchange two rows .
2. Multiply or divide a row by a non zero constant .
3. Add or subtract a multiple of one row from
another.
You can use these three operations to convert the
augmented matrix to a particularly simple form
that al lows you to read o the solutions. The idea
is to eliminate the variables from the equations so
that each variable only occurs once. If there's a
unique solution, that works perfectly . If the system
is indeterminate and has infinitely many solutions,
it doesn't work perfectly, but we'll see how that
works.
You can systematize this elimination process to
form an algorithm. Work from the leftmost column
right one column at a time to simplify the aug-
mented matrix (and, therefore, the system it repre-
sents). The column you're working on is called the
pivot column. Look down the column for the first
nonzero entry that has all 0s to its left. Call that
entry the pivot element, or more simply the pivot.
There are three things to do corresponding to the
three elementary row operations.
1. Exchange the pivot row with the highest row
that has 0s in the pivot column and all columns
to the left. (Do nothing if there is no such row.)
2. Divide the pivot row by the pivot so that the
value of the pivot becomes 1.
3. Subtract multiples of the pivot row from the
other rows to clear out the pivot column (ex-
cept the pivot itself).
When you're all done with this algorithm, the
matrix will be in something called reduced row ech-
elon form. The word echelon comes from a partic-
ular stair step formation of troops. For us, a matrix
is in reduced row echelon form if
1. the rows of all zeros (if any) appear at the bot-
tom of the matrix
2. the first nonzero entry of a nonzero row is 1
3. that leading 1 appears appears to the right of
leading 1s in higher rows
4. all the other entries in a column that has a
leading 1 are 0
If the first three conditions hold, the matrix is said
to be in row echelon form.
The reduction algorithm to convert an aug-
mented matrix to reduced row echelon form goes
by the name Gauss-Jordan reduction. The partial
algorithm that stops with a row echelon form goes
by the name Gaussian elimination. As we've seen,
this reduction was known to the ancient Chinese.
Two martrices are row equivalent if their corre -
sponding systems of linear equations have the same
solutions. It's easy to show that means you can
perform a sequence of elementary row operations
on one to eventually get the other.
Homogeneous systems. A system of linear
equations is called homogeneous if all the constant
terms are 0, that is, it is of the form Ax = 0. Ev-
ery homogeneous system has at least one solution,
namely the trivial solution where all the variables
have the value 0. The question we have for homoge-
neous systems is whether they have any nontrivial
solutions.
Row reduction in Matlab. Section 12.4 of our
text shows how you can perform the elementary row
operations in Matlab, either by explicitly manip-
ulating the rows of a matrix or by using the routine
reduce.
If you're only interested in the resulting reduced
row eschelon form of a matrix, then you can use the
command rref. For example, here it's used for the
system of linear equations
From that reduced row eschelon form, we can de-
termine the general solution. For that y and z may
be freely chosen, and
