Vector Spaces

In this section we present a formal definition of a vector space, which will lead to an extra increment of abstraction. Once defined, we study its most basic properties.

1. Comment on how the vector space $\complex{m}$ went from a theorem () to an example ().
2. 2(3,\,4)+(-6)(1,\,2).
3. Suppose that $\alpha$ is a scalar and $\zerovector$ is the zero vector. Why should we prove anything as obvious as $\alpha\zerovector=\zerovector$ such as we did in ?

The next three problems suggest that under the right situations we can cancel. In practice, these techniques should be avoided in other proofs. Prove each of the following statements.

Here is one of the two most important definitions in the entire course.

Now, there are several important observations to make. Many of these will be easier to understand on a second or third reading, and especially after carefully studying the examples in .

An axiom is often a self-evident truth. Something so fundamental that we all agree it is true and accept it without proof. Typically, it would be the logical underpinning that we would begin to build theorems upon. Some might refer to the ten properties of as axioms, implying that a vector space is a very natural object and the ten properties are the essence of a vector space. We will instead emphasize that we will begin with a definition of a vector space. After studying the remainder of this chapter, you might return here and remind yourself how all our forthcoming theorems and definitions rest on this foundation.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

Our aim in this subsection is to give you a storehouse of examples to work with, to become comfortable with the ten vector space properties and to convince you that the multitude of examples justifies (at least initially) making such a broad definition as . Some of our claims will be justified by reference to previous theorems, we will prove some facts from scratch, and we will do one nontrivial example completely. In other places, our usual thoroughness will be neglected, so grab paper and pencil and play along.

So, the set of all matrices of a fixed size forms a vector space. That entitles us to call a matrix a vector, since a matrix is an element of a vector space. For example, if $A,\,B\in M_{34}$ then we call $A$ and $B$ vectors, and we even use our previous notation for column vectors to refer to $A$ and $B$. So we could legitimately write expressions like \vect{u}+\vect{v}=A+B=B+A=\vect{v}+\vect{u} This could lead to some confusion, but it is not too great a danger. But it is worth comment.

The previous two examples may be less than satisfying. We made all the relevant definitions long ago. And the required verifications were all handled by quoting old theorems. However, it is important to consider these two examples first. We have been studying vectors and matrices carefully (, ), and both objects, along with their operations, have certain properties in common, as you may have noticed in comparing with . Indeed, it is these two theorems that motivate us to formulate the abstract definition of a vector space, . Now, if we prove some general theorems about vector spaces (as we will shortly in ), we can then instantly apply the conclusions to both $\complex{m}$ and $M_{mn}$. Notice too, how we have taken six definitions and two theorems and reduced them down to two examples. With greater generalization and abstraction our old ideas get downgraded in stature.

Let us look at some more examples, now considering some new vector spaces.

Here is a unique example.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

has provided us with an abundance of examples of vector spaces, most of them containing useful and interesting mathematical objects along with natural operations. In this subsection we will prove some general properties of vector spaces. Some of these results will again seem obvious, but it is important to understand why it is necessary to state and prove them. A typical hypothesis will be Let $V$ be a vector space. From this we may assume the ten properties of , and nothing more. It is like starting over, as we learn about what can happen in this new algebra we are learning. But the power of this careful approach is that we can apply these theorems to any vector space we encounter those in the previous examples, or new ones we have not yet contemplated. Or perhaps new ones that nobody has ever contemplated. We will illustrate some of these results with examples from the crazy vector space (), but mostly we are stating theorems and doing proofs. These proofs do not get too involved, but are not trivial either, so these are good theorems to try proving yourself before you study the proof given here. (See .)

First we show that there is just one zero vector. Notice that the properties only require there to be at least one, and say nothing about there possibly being more. That is because we can use the ten properties of a vector space () to learn that there can never be more than one. To require that this extra condition be stated as an eleventh property would make the definition of a vector space more complicated than it needs to be.

Condensed — the full section is in Beezer, A First Course in Linear Algebra.

When we say that $V$ is a vector space, we then know we have a set of objects (the vectors), but we also know we have been provided with two operations (vector addition and scalar multiplication) and these operations behave with these objects according to the ten properties of . One combines two vectors and produces a vector, the other takes a scalar and a vector, producing a vector as the result. So if $\vect{u}_1,\,\vect{u}_2,\,\vect{u}_3\in V$ then an expression like 5\vect{u}_1+7\vect{u}_2-13\vect{u}_3 would be unambiguous in any of the vector spaces we have discussed in this section. And the resulting object would be another vector in the vector space. If you were tempted to call the above expression a linear combination, you would be right. Four of the definitions that were central to our discussions in were stated in the context of vectors being column vectors, but were purposely kept broad enough that they could be applied in the context of any vector space. They only rely on the presence of scalars, vectors, vector addition and scalar multiplication to make sense. We will restate them shortly, unchanged, except that their titles and acronyms no longer refer to column vectors, and the hypothesis of being in a vector space has been added. Take the time now to look forward and review each one, and begin to form some connections to what we have done earlier and what we will be doing in subsequent sections and chapters. Specifically, compare the following pairs of definitions: