Page 113 - A Course in Linear Algebra with Applications

4.2: Vector Spaces and Subspaces 97

(
Hence -l)v = —v by (b).
Subspaces

Roughly speaking, a subspace is a vector space contained
within a larger vector space; for example, the vector space
p2(R) is a subspace of Ps(R). More precisely, a subset S of
a vector space V is called a subspace of V if the following
statements are true:

(i) S contains the zero vector 0;
(ii) if v belongs to S, then so does cv for every scalar c,
that is, S is closed under scalar multiplication;
(iii) if u and v belong to S, then so does u + v that is, S
is closed under addition.

Thus a subspace of V is a subset S which is itself a vector
space with respect to the same rules of addition and scalar
multiplication as V. Of course, the vector space axioms hold
in S since they are already valid in V.

Examples of subspaces
If V is any vector space, then V itself is a subspace, for
trivial reasons. It is often called the improper subspace. At
the other extreme is the zero subspace, written 0 or Oy, which
contains only the zero vector 0. This is the smallest subspace
of V. (In general a vector space that contains only the zero
vector is called a zero space). The zero subspace and the
improper subspace are present in every vector space. We move
on now to some more interesting examples of subspaces.

Example 4.2.1
2
Let S be the subset of R consisting of all columns of the form

(-30