A few comments are in order on our axioms and notation. There are striking similarities (and equally striking differences) between the axioms for a field and the axioms for a vector space over a field. In both cases, the axioms (A) describe the additive structure of the system, the axioms (B) describe its multiplicative structure, and the axioms (C) describe the connection between the two structures. Those familiar with algebraic terminology will have recognized the axioms (A) (in both Sections 1 and 2) as the defining conditions of an abelian (commutative) group; the axioms (B) and (C) (in Section: Vector spaces ) express the fact that the group admits scalars as operators. We mention in passing that if the scalars are elements of a ring (instead of a field), the generalized concept corresponding to a vector space is called a module .

Special real vector spaces (such as \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\) ) are familiar in geometry. There seems at this stage to be no excuse for our apparently uninteresting insistence on fields other than \(\mathbb{R}\) , and, in particular, on the field \(\mathbb{C}\) of complex numbers. We hope that the reader is willing to take it on faith that we shall have to make use of deep properties of complex numbers later (conjugation, algebraic closure), and that in both the applications of vector spaces to modern (quantum mechanical) physics and the mathematical generalization of our results to Hilbert space, complex numbers play an important role. Their one great disadvantage is the difficulty of drawing pictures; the ordinary picture (Argand diagram) of \(\mathbb{C}^{1}\) is indistinguishable from that of \(\mathbb{R}^{2}\) , and a graphic representation of \(\mathbb{C}^{2}\) seems to be out of human reach. On the occasions when we have to use pictorial language we shall therefore use the terminology of \(\mathbb{R}^{n}\) in \(\mathbb{C}^{n}\) , and speak of \(\mathbb{C}^{2}\) , for example, as a plane.

Finally we comment on notation. We observe that the symbol \(0\) has been used in two meanings: once as a scalar and once as a vector. To make the situation worse, we shall later, when we introduce linear functionals and linear transformations, give it still other meanings. Fortunately the relations among the various interpretations of \(0\) are such that, after this word of warning, no confusion should arise from this practice.

EXERCISES

Exercise 1. Prove that if \(x\) and \(y\) are vectors and if \(\alpha\) is a scalar, then the following relations hold.

  1. \(0+x=x\) .
  2. \(-0=0\) .
  3. \(\alpha \cdot 0=0\) .
  4. \(0 \cdot x=0\) . (Observe that the same symbol is used on both sides of this equation; on the left it denotes a scalar, on the right it denotes a vector.)
  5. If \(\alpha x=0\) , then either \(\alpha=0\) or \(x=0\) (or both).
  6. \(-x=(-1) x\) .
  7. \(y+(x-y)=x\) . (Here \(x-y=x+(-y)\) .)

Exercise 2. If \(p\) is a prime, then \(\mathbb{Z}_{p}^{n}\) is a vector space over \(\mathbb{Z}_{p}\) (cf. Section: Fields , Ex. 3); how many vectors are there in this vector space?

Exercise 3. Let \(\mathcal{V}\) be the set of all (ordered) pairs of real numbers. If \(x=(\xi_{1}, \xi_{2})\) and \(y=(\eta_{1}, \eta_{2})\) are elements of \(\mathcal{V}\) , write \begin{align} x+y & =(\xi_{1}+\eta_{1}, \xi_{2}+\eta_{2}) \\ \alpha x & =(\alpha \xi_{1}, 0) \\ 0 & =(0,0) \\ -x & =(-\xi_{1},-\xi_{2}). \end{align} 

Is \(\mathcal{V}\) a vector space with respect to these definitions of the linear operations? Why?

Exercise 4. Sometimes a subset of a vector space is itself a vector space (with respect to the linear operations already given). Consider, for example, the vector space \(\mathbb{C}^{3}\) and the subsets \(\mathcal{V}\) of \(\mathbb{C}^{3}\) consisting of those vectors \((\xi_{1}, \xi_{2}, \xi_{3})\) for which

  1. \(\xi_{1}\) is real,
  2. \(\xi_{1}=0\) ,
  3. either \(\xi_{1}=0\) or \(\xi_{2}=0\) ,
  4. \(\xi_{1}+\xi_{2}=0\) ,
  5. \(\xi_{1}+\xi_{2}=1\) .

In which of these cases is \(\mathcal{V}\) a vector space?

Exercise 5. Consider the vector space \(\mathcal{P}\) and the subsets \(\mathcal{V}\) of \(\mathcal{P}\) consisting of those vectors (polynomials) \(x\) for which

  1. \(x\) has degree \(3\) ,
  2. \(2 x(0)=x(1)\) ,
  3. \(x(t) \geq 0\) whenever \(0 \leq t \leq 1\) ,
  4. \(x(t)=x(1-t)\) for all \(t\) .

In which of these cases is \(\mathcal{V}\) a vector space?