Math 1b Prac — Rowspaces, nullspaces, orthogonal spaces
January 23, 2009
[As usual, the notes I post on the web are sketchy and more of an outline. Some more
details in class.]
Subspaces of
R
n
arise, as an oversimpliFcation, in two ways: either as the span of
some vectors or as the solutions of a homogeneous system of linear equations. It is useful
to be able to go from one kind of description to the other.
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
nullspace
(or solution space) of
M
is the set of all column
vectors
x
so that
M
x
=
0
. 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
.
Given a subspace
U
of
R
n
orthogonal space
U
⊥
(read:
U
perp)istheseto
fa
l
l
vectors
y
so that
x
·
y
=0fora
l
l
x
∈
U
. To check this condition (
x
·
y
l
l
x
∈
U
),
it is suﬃcient to check that
y
is orthogonal to a spanning set of
U
, so assuming we have a
spanning set
u
1
,
u
2
,...,
u
k
for
U
, a vector
y
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 Winter '09
 Bopanna
 Math, Linear Algebra, Vector Space, UK, rows

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