92 CHAPTER 4. LINEAR SPACES

the existence of an additive inverse, Also

α (v+w) = α v+αw, (4.7)

(α +β ) v = α v+βv, (4.8)

α (βv) = αβ (v) , (4.9)

1v = v. (4.10)

In the above 0= (0, · · · ,0).

You should verify these properties all hold. For example, consider 4.7

α (v+w) = α (v1 +w1, · · · ,vn +wn)

= (α (v1 +w1) , · · · ,α (vn +wn))

= (αv1 +αw1, · · · ,αvn +αwn)

= (αv1, · · · ,αvn)+(αw1, · · · ,αwn)

= αv+αw.

As usual subtraction is defined as x−y ≡ x+(−y) .

4.2 Subspaces Spans and BasesAs mentioned above, Fn is an example of a vector space. In dealing with vector spaces,the concept of linear combination is fundamental. When one considers only algebraicconsiderations, it makes no difference what field of scalars you are using. It could be R, C,Q or even a field of residue classes. However, go ahead and think R or C since the subjectof interest here is analysis.

Definition 4.2.1 Let{x1, · · · ,xp

}be vectors in a vector space Y having the field

of scalars F. A linear combination is any expression of the form ∑pi=1 cixi where the ci are

scalars. The set of all linear combinations of these vectors is called span(x1, · · · ,xp) . Avector v is said to be in the span of some set S of vectors if v is a linear combination ofvectors of S. This means: finite linear combination. If V ⊆Y, then V is called a subspaceif it contains 0 and whenever α,β are scalars and u and v are vectors of V, it followsαu+βv ∈V . That is, it is “closed under the algebraic operations of vector addition andscalar multiplication” and is therefore, a vector space. A linear combination of vectorsis said to be trivial if all the scalars in the linear combination equal zero. A set of vectorsis said to be linearly independent if the only linear combination of these vectors whichequals the zero vector is the trivial linear combination. Thus {x1, · · · ,xn} is called linearlyindependent if whenever ∑

nk=1 ckxk = 0, it follows that all the scalars, ck equal zero. A set

of vectors, {x1, · · · ,xn} , is called linearly dependent if it is not linearly independent. Thusthe set of vectors is linearly dependent if there exist scalars, ci, i = 1, · · · ,n, not all zerosuch that ∑

nk=1 ckxk = 0.

Lemma 4.2.2 A set of vectors {x1, · · · ,xn} is linearly independent if and only if noneof the vectors can be obtained as a linear combination of the others.