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Constantin Teleman, Math 54.1, Fall 09
1. Notation and refreshers
Unless otherwise speciﬁed,
A
is an
m
×
n
real matrix, and vectors
x
are column vectors. Row vectors
are written as
x
T
, where
x
is the corresponding column vector. The nullspace of
A
, Null(
A
)
⊂
R
n
,
is the space of solutions of the homogeneous system
A
x
=
0
, the column space Col(
A
)
⊂
R
m
is the span of the columns. There are two more subspaces associated to
A
, this time consisting
of
row vectors
: the row space Row(
A
)
⊂
R
n
, the span of the rows in
A
; and the left nullspace
LNull(
A
)
⊂
R
m
that we’ll meet below. (A quick and dirty deﬁnition is as the nullspace of the
‘ﬂipped’ or transposed matrix
A
T
: this is the matrix with switched indexing, (
A
T
)
ij
=
A
ji
.)
Caution!
These four subspaces are distinct in general, with
no obvious
relation among them.
We will learn some subtle relations later.
The reduced row echelon form of
A
is the matrix rref(
A
) produced from elementary row oper
ations, with the properties that
•
All zerorows are at the bottom
•
Every row which is not all zero starts with a 1, called the pivot or leading 1,
•
Every pivot is strictly to the right of all pivots in the rows above it, and
•
All entries above (and below) a pivot are zero.
The columns containing pivots are called
pivot columns
, the others are the
free columns
. The
nullspace of
A
agrees with that of rref(
A
), and can be parametrized as follows: the free variables
can be chosen freely, and each equation in the reduced row system can then be used to solve for
the corresponding pivot variable.
A similar story applies to the inhomogeneous system
A
x
=
b
: it has the same general solution
as the reduced system rref(
A
)
x
=
b
0
, where [rref(
A
)

b
0
] is the reduced form of the augmented
matrix [
A

b
]. The general solution can be parametrized by the same procedure.
We will learn a
strong uniqueness property
of rref(
A
): it is completely determined by nullspace
of
A
. (Similarly, the reduced form of the augmented matrix [
A

b
] is determined uniquely from the
aﬃne space of solutions of
A
x
=
b
.)
A collection of vectors
{
v
1
,...,
v
r
}
is
linearly independent
if the only expression of
0
as a linear
combination of the
v
i
is the one with 0 weights: that is,
k
z
v
1
+
···
+
k
r
v
r
=
0
⇒
k
1
=
k
2
=
···
=
k
r
= 0
.
A collection
{
w
1
,...,
w
s
}
spans
the linear subspace
L
∈
R
n
if every
w
i
lies in
L
and every vector
in
L
can be expressed as a linear combination of the
w
i
. A linearly independent, ordered collection
of vectors which spans
L
is called a
basis
. Each vector in
L
can be
uniquely
expressed as a linear
combination of the basis elements (that is, the weights are uniquely determined). The main example
is the
standard basis
e
1
,...,
e
n
of
R
n
, the unit vectors on the coordinate axes.
1
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This note was uploaded on 12/01/2009 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at University of California, Berkeley.
 Spring '08
 Chorin
 Vectors

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