Page 167 - Matrix Analysis & Applied Linear Algebra

162 Chapter 4 Vector Spaces

Example 4.1.4
2
Consider the vector space , and let

L = {(x, y) | y = αx}

2
be a line through the origin. L is a subset of , but L is a special kind
of subset because L also satisfies the properties (A1)–(A5) and (M1)–(M5)
that define a vector space. This shows that it is possible for one vector space to
properly contain other vector spaces.

Subspaces

Let S be a nonempty subset of a vector space V over F (symbolically,
S⊆ V). If S is also a vector space over F using the same addition
and scalar multiplication operations, then S is said to be a subspace of
V. It’s not necessary to check all 10 of the defining conditions in order
to determine if a subset is also a subspace—only the closure conditions
(A1) and (M1) need to be considered. That is, a nonempty subset S
of a vector space V is a subspace of V if and only if
(A1) x, y ∈S =⇒ x + y ∈S
and
(M1) x ∈S =⇒ αx ∈S for all α ∈F.

Proof. If S is a subset of V, then S automatically inherits all of the vector
space properties of V except (A1), (A4), (A5), and (M1). However, (A1)
together with (M1) implies (A4) and (A5). To prove this, observe that (M1)
implies (−x)=(−1)x ∈S for all x ∈S so that (A5) holds. Since x and (−x)
are now both in S, (A1) insures that x +(−x) ∈S, and thus 0 ∈S.

Example 4.1.5
Given a vector space V, the set Z = {0} containing only the zero vector is
a subspace of V because (A1) and (M1) are trivially satisfied. Naturally, this
subspace is called the trivial subspace.

3
2
Vector addition in and is easily visualized by using the parallelo-
gram law, which states that for two vectors u and v, the sum u + v is the
vector defined by the diagonal of the parallelogram as shown in Figure 4.1.1.