As an application of the notion of linear basis, or coordinate system, we shall now fulfill an implicit earlier promise by showing that every finite-dimensional vector space over a field \(\mathbb{F}\) is essentially the same as (in technical language, is isomorphic to) some \(\mathbb{F}^{n}\) .
Definition 1. Two vector spaces \(\mathcal{U}\) and \(\mathcal{V}\) (over the same field) are isomorphic if there is a one-to-one correspondence between the vectors \(x\) of \(\mathcal{U}\) and the vectors \(y\) of \(\mathcal{V}\) , say \(y=T(x)\) , such that \[T(\alpha_{1} x_{1}+\alpha_{2} x_{2})=\alpha_{1} T(x_{1})+\alpha_{2} T(x_{2})\] In other words, \(\mathcal{U}\) and \(\mathcal{V}\) are isomorphic if there is an isomorphism (such as \(T\) ) between them, where an isomorphism is a one-to-one correspondence that preserves all linear relations.
It is easy to see that isomorphic finite-dimensional vector spaces have the same dimension; to each basis in one space there corresponds a basis in the other space. Thus dimension is an isomorphism invariant; we shall now show that it is the only isomorphism invariant, in the sense that every two vector spaces with the same finite dimension (over the same field, of course) are isomorphic. Since the isomorphism of \(\mathcal{U}\) and \(\mathcal{V}\) on the one hand, and of \(\mathcal{V}\) and \(\mathcal{W}\) on the other hand, implies that \(\mathcal{U}\) and \(\mathcal{W}\) are isomorphic, it will be sufficient to prove the following theorem.
Theorem 1. Every \(n\) -dimensional vector space \(\mathcal{V}\) over a field \(\mathbb{F}\) is isomorphic to \(\mathbb{F}^{n}\) .
Proof. Let \(\{x_{1}, \ldots, x_{n}\}\) be any basis in \(\mathcal{V}\) . Each \(x\) in \(\mathcal{V}\) can be written in the form \(\xi_{1} x_{1}+\cdots+\xi_{n} x_{n}\) , and we know that the scalars \(\xi_{1}, \ldots, \xi_{n}\) are uniquely determined by \(x\) . We consider the one-to-one correspondence \[x \rightleftarrows(\xi_{1}, \ldots, \xi_{n})\] between \(\mathcal{V}\) and \(\mathbb{F}^{n}\) . If \(y=\eta_{1} x_{1}+\cdots+\eta_{n} x_{n}\) , then \[\alpha x+\beta y=(\alpha \xi_{1}+\beta \eta_{1}) x_{1}+\cdots+(\alpha \xi_{n}+\beta \eta_{n}) x_{n}\] this establishes the desired isomorphism. ◻
One might be tempted to say that from now on it would be silly to try to preserve an appearance of generality by talking of the general \(n\) -dimensional vector space, since we know that, from the point of view of studying linear problems, isomorphic vector spaces are indistinguishable, and, consequently, we might as well always study \(\mathbb{F}^{n}\) . There is one catch. The most important properties of vectors and vector spaces are the ones that are independent of coordinate systems, or, in other words, the ones that are invariant under isomorphisms. The correspondence between \(\mathcal{V}\) and \(\mathbb{F}^{n}\) was, however, established by choosing a coordinate system; were we always to study \(\mathbb{F}^{n}\) , we would always be tied down to that particular coordinate system, or else we would always be faced with the chore of showing that our definitions and theorems are independent of the coordinate system in which they happen to be stated. (This horrible dilemma will become clear later, on the few occasions when we shall be forced to use a particular coordinate system to give a definition.) Accordingly, in the greater part of this book, we shall ignore the theorem just proved, and we shall treat \(n\) -dimensional vector spaces as self-respecting entities, independently of any basis. Besides the reasons just mentioned, there is another reason for doing this: many special examples of vector spaces, such for instance as \(\mathcal{P}_{n}\) , would lose a lot of their intuitive content if we were to transform them into \(\mathbb{C}^{n}\) and speak of coordinates only. In studying vector spaces, such as \(\mathcal{P}_{n}\) , and their relation to other vector spaces, we must be able to handle them with equal ease in different coordinate systems, or, and this is essentially the same thing, we must be able to handle them without using any coordinate systems at all.
EXRRCISES
Exercise 1.
- What is the dimension of the set \(\mathbb{C}\) of all complex numbers considered as a real vector space? (See Section: Examples , (9).)
- Every complex vector space \(\mathcal{V}\) is intimately associated with a real vector space \(\mathcal{V}^{-}\) ; the space \(\mathcal{V}^{-}\) is obtained from \(\mathcal{V}\) by refusing to multiply vectors of \(\mathcal{V}\) by anything other than real scalars. If the dimension of the complex vector space \(\mathcal{V}\) is \(n\) , what is the dimension of the real vector space \(\mathcal{V}^{-}\) ?
Exercise 2. Is the set \(\mathbb{R}\) of all real numbers a finite-dimensional vector space over the field \(\mathbb{Q}\) of all rational numbers? (See Section: Examples , (8). The question is not trivial; it helps to know something about cardinal numbers.)
Exercise 3. How many vectors are there in an \(n\) -dimensional vector space over the field \(\mathbb{Z}_{p}\) (where \(p\) is a prime)?
Exercise 4. Discuss the following assertion: if two rational vector spaces have the same cardinal number (i.e., if there is some one-to-one correspondence between them), then they are isomorphic (i.e., there is a linearity-preserving one-to-one correspondence between them). A knowledge of the basic facts of cardinal arithmetic is needed for an intelligent discussion.