Chapter 4: The Vector Space
R
n
4.1: Subspaces of
R
n
and Spanning Sets
Introduction
In Chapter 4 we move from
R
3
to
R
n
,
which immediately makes
things more abstract, since you can’t visualize higher dimensions.
In addition, Chapter 4 introduces many new concepts: subspace,
spanning sets, span of a set, linear independence, linear
dependence, basis, dimension, null space, row space, column space,
orthogonal complements, orthonormal basis, orthogonal matrix, .
..
to mention but a few! The good news is that there are no new
computational techniques involved. Everything we shall do in
Chapter 4 comes down to either solving a system of equations,
especially homogeneous systems of equations; reducing a matrix;
ﬁnding a determinant; adding or subtracting vectors; multiplying
vectors by a scalar; multiplying matrices; or calculating dot
products. There are welldeﬁned algorithms for everything we shall
do. It’s just that sometimes you may lose sight of why you are
doing certain calculations. The payoﬀ will be some very important
results, cf. Sections 4.6 and 4.7.
Chapter 4 Lecture Notes
MAT188H1F Lec03 Burbulla
Chapter 4: The Vector Space
R
n
4.1: Subspaces of
R
n
and Spanning Sets
Set Notation
Since we will be talking a lot about sets of vectors, here’s a quick
review (?) of set theory notation.
I
x
∈
A
is read “
x
is an element of the set
A
,
” or “
x
is in
A
.
”
I
A
=
{
x
,
y
,
z
, . . .
}
is a list of the elements in
A
.
I
A
=
{
x
∈
B

P
(
x
) is true
}
is read “
A
is the set of
x
in
B
such
that the property
P
(
x
) is true.”
I
φ
is the empty set, the set with no elements.
I
For example, in calculus, [
a
,
b
] =
{
x
∈
R

a
≤
x
≤
b
}
is set
notation for the closed interval [
a
,
b
]; it means “the set of
x
in
R
such that
a
≤
x
≤
b
.
”
I
For example, in linear algebra,
{
X
∈
R
3

AX
=
B
}
is the set of
solutions in
R
3
to the system of linear equations
AX
=
B
.
Chapter 4 Lecture Notes
MAT188H1F Lec03 Burbulla