Linear Algebra

94 Chapter 2. Vector Spaces
second. For the fourth, consider the zero vector of V and note that closure of S
under linear combinations of pairs of vectors gives that (where �s is any member
of the nonempty set S) 0· �s +0· �s = �0 isinS; showing that �0 acts under the
inherited operations as the additive identity of S is easy. The fifth condition is
satisfied because for any �s ∈ S, closure under linear combinations shows that
the vector 0 · �0 +(−1) · �s is in S; showing that it is the additive inverse of �s
under the inherited operations is routine.
The checks for item (2) are similar and are saved for Exercise 32. QED
We usually show that a subset is a subspace with (2) =⇒ (1).
2.10 Remark At the start of this chapter we introduced vector spaces as collections
in which linear combinations are “sensible”. The above result speaks
to this.
The vector space definition has ten conditions but eight of them, the ones
stated there with the ‘•’ bullets, simply ensure that referring to the operations
as an ‘addition’ and a ‘scalar multiplication’ is sensible. The proof above checks
that if the nonempty set S satisfies statement (2) then inheritance of the operations
from the surrounding vector space brings with it the inheritance of these
eight properties also (i.e., commutativity of addition in S follows right from
commutativity of addition in V ). So, in this context, this meaning of “sensible”
is automatically satisfied.
In assuring us that this first meaning of the word is met, the result draws
our attention to the second meaning. It has to do with the two remaining
conditions, the closure conditions. Above, the two separate closure conditions
inherent in statement (1) are combined in statement (2) into the single condition
of closure under all linear combinations of two vectors, which is then extended
in statement (3) to closure under combinations of any number of vectors. The
latter two statements say that we can always make sense of an expression like
r1�s1 + r2�s2, without restrictions on the r’s — such expressions are “sensible” in
that the vector described is defined and is in the set S.
This second meaning suggests that a good way to think of a vector space
is as a collection of unrestricted linear combinations. The next two examples
take some spaces and describe them in this way. That is, in these examples we
paramatrize, just as we did in Chapter One to describe the solution set of a
homogeneous linear system.
2.11 Example This subset of R 3
⎛
S = { ⎝ x
⎞
y⎠
z
� � x − 2y + z =0}
is a subspace under the usual addition and scalar multiplication operations of
column vectors (the check that it is nonempty and closed under linear combinations
of two vectors is just like the one in Example 2.2). To paramatrize, we