Math 1b Prac — Bases for row spaces, null spaces; orthogonal spaces
January 18, 2011 — slightly revised January 21, 2011
Given a matrix
M
, the
row space
of
M
is the span of its rows (the set of all linear
combinations of its rows). The
null space
of
M
is the set of all column vectors
x
so that
M
x
=
0
. In other words, the null space of
M
is the set of all vectors orthogonal to the rows
of
M
. In yet other words, the null space of
M
is the “solution space” of the homogeneous
system of linear equations corresponding to
M
. We will check (it is not hard) that these
are really subspaces. Row operations on a matrix
M
do not change the row space or the
nullspace of
M
.
These notes are very concise. Proofs of the following theorems will be given in class,
or possibly in an expanded version of this handout.
We use the matrix
M
=
⎛
⎝
1
2
3
4
2
3
4
5
3
4
5
6
⎞
⎠
in our examples.
∗
∗
∗
Two ways to find a basis for the row space of
A
Theorem 1.
Let
E
be a basic form of
A
. Then the nonzero rows of
E
form a basis for
the row space of
A
.
Example: The reduced echelon form of
M
is
E
=
⎛
⎝
1
0
−
1
−
2
0
1
2
3
0
0
0
0
⎞
⎠
.
So one basis for the row space of
M
is, or consists of, (1
,
0
,
−
1
,
−
2) and (0
,
1
,
2
,
3).
Theorem 2.
Let
E
be a basic form of
A
. Then the the columns of
A
that correspond to
the special columns of
E
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 Winter '08
 Aschbacher
 Math, Linear Algebra, UK, row space, 12 m

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