(1)
System of linear equations
:
m
linear equations
∑
n
j
=1
a
ij
x
j
=
b
i
for
i
= 1
,...,m
in
n
unknowns
x
1
,...,x
n
; called
homogeneous
if all
b
i
= 0.
A
= [
a
ij
] is the
matrix of
coeﬃcients
, [
A
k
b
] is the
augmented matrix
obtained by adjoining the
stub
b
=
•
b
1
.
.
.
b
m
‚
.
For homogeneous systems we usually omit the stub (it remains constantly zero throughout
the reduction process). A
solution
of a system is a value (
x
1
,...,x
n
) = (
a
1
,...,a
n
) for
the variables which satisﬁes all equations of the system
simultaneously
. A system either has
no solutions
(
inconsistent system
), a
single solution
(
unique system
) [
independent
of
any further parameters], or has
inﬁnitely many solutions
(
inﬁnite system
[
dependent
on a
choice of free parameters]. IT IS RECOMMENDED THAT YOU
ALWAYS
WRITE THE
VARIABLES ABOVE THEIR COLUMNS IN THE MATRIX OF COEFFICIENTS
A
in a
system
AX
=
B
, AND SEPARATE THE STUB
B
FROM THE MATRIX OF COEFFI
CIENTS BY A DOUBLE BAR
k
(reminding you of an upright elongated equal sign), for
example
x
1
x
2
x
3
2

1
3
5
6
8
2
9
3
0
5
7
. The augmented matrix may have a generic stub whose entries are variables
b
i
or
y
i
instead
of constants (when you are solving
Ax
=
y
for all possible
y
’s at once), or it may contain
several numerical columns
b
i
(when you are simultaneously solving several systems
Ax
=
b
i
with the same
A
).
SKILL
: be able to translate back and forth between a system of explicit equations and its
augmented matrix. A
shmoo system
presents the answer to you directly (the augmented
matrix is in reduced rowechelon form).
(2)
Elementary row operations
E
on matrices: (I)
E
(
i,j
)
swaps
(interchanges) the
i
th and
j
th rows, (II)
E
(
i,c
)
scales
or multiplies the
i
th row by nonzero constant
c
, (III)
E
(
i,c,j
)
(
addmul
) multiplies the
j
th row by
c
and adds it to the
i
th . Fact: The elementary
operations are reversible: the reverse of swap is swap, the reverse of scale by
c
is scale
by the reciprocal 1
/c
, the reverse of addmul by
c
is addmul by the negative

c
(add

c
times the original row to the altered one).
Thus these operations on rows do not change the
mathematical content of the system (the solution set), they merely changes the packaging.
[Beware: the analogous elementary operation on columns DO change the mathematical
content, since they change the variables; the variables are encoded in the augmented matrix
by positional notation as labels for the columns, and these must not be mixed or altered.
This is the reason it is useful to label the columns with their variables, which labeling stays
the same throughout the reduction process.]
(3)
Echelon
and
Reduced rowechelon form
: A matrix is in
[row]echelon form
(ref)
iﬀ (i) all zero rows are at the bottom, (ii) all nonzero rows begin with a
leading one
,
(iii) the leading ones appear in
echelon
(stairstep) form, descending downward and to the
right. This implies (iv)
0
the entries below any leading one are all zero (the following rows are