Linear Algebra, 2020a

92 Chapter Two. Vector Spaces
Another answer is perhaps more satisfying. People in this area have worked
to develop the right balance of power and generality. This definition is shaped
so that it contains the conditions needed to prove all of the interesting and
important properties of spaces of linear combinations. As we proceed, we shall
derive all of the properties natural to collections of linear combinations from the
conditions given in the definition.
The next result is an example. We do not need to include these properties
in the definition of vector space because they follow from the properties already
listed there.
1.16 Lemma In any vector space V, for any ⃗v ∈ V and r ∈ R, we have (1) 0·⃗v = ⃗0,
(2) (−1 · ⃗v)+⃗v = ⃗0, and (3) r · ⃗0 = ⃗0.
Proof For (1) note that ⃗v =(1 + 0) · ⃗v = ⃗v +(0 · ⃗v). Add to both sides the
additive inverse of ⃗v, the vector ⃗w such that ⃗w + ⃗v = ⃗0.
⃗w + ⃗v = ⃗w + ⃗v + 0 · ⃗v
⃗0 = ⃗0 + 0 · ⃗v
⃗0 = 0 · ⃗v
Item (2) is easy: (−1 ·⃗v)+⃗v =(−1 + 1) ·⃗v = 0 ·⃗v = ⃗0. For (3), r ·⃗0 = r · (0 ·⃗0) =
(r · 0) · ⃗0 = ⃗0 will do. QED
The second item shows that we can write the additive inverse of ⃗v as ‘−⃗v ’
without worrying about any confusion with (−1) · ⃗v.
A recap: our study in Chapter One of Gaussian reduction led us to consider
collections of linear combinations. So in this chapter we have defined a vector
space to be a structure in which we can form such combinations, subject to
simple conditions on the addition and scalar multiplication operations. In a
phrase: vector spaces are the right context in which to study linearity.
From the fact that it forms a whole chapter, and especially because that
chapter is the first one, a reader could suppose that our purpose in this book is
the study of linear systems. The truth is that we will not so much use vector
spaces in the study of linear systems as we instead have linear systems start us
on the study of vector spaces. The wide variety of examples from this subsection
shows that the study of vector spaces is interesting and important in its own
right. Linear systems won’t go away. But from now on our primary objects of
study will be vector spaces.
Exercises
1.17 Name the zero vector for each of these vector spaces.
(a) The space of degree three polynomials under the natural operations.