08 Vector Spaces
We have been thinking of a "vector" as being a column, or sometimes a row, of numbers. These are "vectors in $\R^n$." In Chapter 2, we move to a more abstract view, where a vector is simply an element of something called a "vector space." The discussion will become more abstract and more theoretical, and there will be more proofs. However, in the end, most of the practical work is still done using matrices and columns of numbers, and you should still tend to visualize vectors as arrows in $\R^2$ or $\R^3$ when you what to get a more intuitive understanding of what is going on.
Definition: A vector space $(V,+,\cdot)$ over $\R$ is defined to be a set, $V$, together with two binary operations $+$ and $\cdot$, which are called vector addition and scalar multiplication. Binary operation means that the operator takes two inputs and produces some output. Vector addition takes two elements of $V$ as input. Scalar multiplication takes a scalar, that is a real number, and an element of $V$ as input. Vector addition and scalar multiplication must satisfy ten properties:
- $\vec v+\vec u\in V$ for all $\vec v,\vec u\in V$ (closure under vector addition)
- $\vec v+\vec u=\vec u+\vec v$ for all $\vec v,\vec u\in V$ (vector addition is commutative)
- $\vec v+(\vec u+\vec w)=(\vec u+\vec v)+\vec w$ for all $\vec v,\vec u,\vec w\in V$ (vector addition is associative)
- there is a $\vec 0\in V$ such that $\vec v + \vec 0 = \vec v$ for all $\vec v\in V$ (additive identity)
- for any $\vec v\in V$, there is $-\vec v\in V$ such that $\vec v + (-\vec v)=\vec 0$ (additive inverse)
- $r\cdot\vec v \in V$ for all $r\in\R$ and $\vec v\in V$ (closure under scalar multiplication)
- $(r+s)\cdot \vec v = (r\cdot \vec v) + (s\cdot\vec v)$ for all $r,s\in\R$ and $\vec v\in V$ (first distributive property)
- $r\cdot(\vec v +\vec u) = (r\cdot\vec v)+(r\cdot\vec u)$ for all $r\in\R$ and $\vec v,\vec u\in V$ (second distributive property)
- $(rs)\cdot\vec v = r\cdot (s\cdot\vec v)$ for all $r,s\in\R$ and $\vec v\in V$ (second associative property)
- $1\cdot\vec v = \vec v$ for all $\vec v\in V$ (multiplicative identity)
These properties are all things that you already know are true, or that are obvious, for the usual column vectors in $\R^n$. You certainly don't have to remember the number for each property, but you should know that they are true for any vector space, and you should be ready to pull them out when you need them in a proof or calculation.
The vector space $\R^n$ of column vectors is still our most basic and important example. We will see that in some sense, any so-called "finite dimensional" vector space is really the same as $\R^n.$ The technical term is that it is isomorphic to $\R^n$ as a vector space. There are also "infinite dimensional" vector spaces, but we will mostly avoid them except for some examples.
Chapter One, Section 1.I works through many examples of vector spaces, and you should pay attention to them. The most important, since we will often use them in future examples, are $\mathcal P_n$, for $n\in \N$, and the vector space that I will call $\mathcal F$:
$\mathcal P_n$ is the set of polynomials with real coefficients and degree less than or equal to $n$. Vector addition in this vector space is the usual addition of polynomials and scalar multiplication is the usual multiplication of a polynomial by a constant. That is,
- $\mathcal P_n=\{a_o+a_1x+a_2x^2+\cdots+a_nx^n\,|\,a_o,a_1,\dots,a_n\in\R\}$
- $(a_o+a_1x+a_2x^2+\cdots+a_nx^n) + (b_o+b_1x+b_2x^2+\cdots+b_nx^n)$ =
$(a_o+b_o)+(a_1+b_1)x+(a_2+b_2)x^2+\cdots+(a_n+b_n)x^n$ - $r\cdot(a_o+a_1x+a_2x^2+\cdots+a_nx^n) = (ra_o)+(ra_1)x+(ra_2)x^2+\cdots+(ra_n)x^n$
We also consider the space $\mathcal P$ of all polynomials, of any degree. The vector space $\mathcal P$ is an example of an infinite-dimensional vector space.
$\mathscr F$ is the set of real-valued functions of a real variable, with the usual addition of functions and the usual multiplication of a scalar times a function. That is,
- $\mathscr F = \{ f\,|\, f:\R\to\R\}$
- $f+g$ is defined by $(f+g)(x)=f(x)+g(x)$ for all $x\in\R$
- $r\cdot f$ is defined by $(r\cdot f)(x) = r(f(x))$ for all $x\in\R$
$\mathscr F$ is an example of a function space. There are many other function spaces. For example, $\mathscr C$, the vector space of continuous functions, or $\mathscr D$, the vector space of differentiable functions. Function spaces such as these are among the most important vector spaces in some branches of mathematics. We could also look at real-valued functions defined on other sets. The book uses the set of functions from $\N$ to $\R$ as an example. The set of functions on the closed interval $[a,b]$ would be another example. In all of these cases, vector addition and scalar multiplication are defined in the same way.
The ten properties that are listed in the definition of vector space are not the only properties that are true for all vector spaces. But they represent a set of axioms for vector spaces. The axioms are assumed to be true of all vector spaces. They can be used to prove other facts about vector spaces.
For example, it is not assumed that $0\cdot \vec v=\vec 0$ for all vectors $\vec v$, but it can be proved using only the axioms and facts about real numbers: $$\begin{align*} 0\cdot\vec v &=(0+0)\cdot \vec v&\mbox{(property of $0\in\R$)}\\ 0\cdot\vec v &=(0\cdot \vec v)+(0\cdot \vec v)&\mbox{(distributive law)} \end{align*}$$ By the additive inverse property, $0\cdot\vec v$ has an additive inverse $-(0\cdot\vec v)$. Add it to both sides of the above equation: $$\begin{align*} 0\cdot\vec v + (-(0\cdot\vec v))&=((0\cdot \vec v)+(0\cdot \vec v)) +(-(0\cdot\vec v))\\ 0\cdot\vec v + (-(0\cdot\vec v))&=(0\cdot \vec v)+((0\cdot \vec v) +(-(0\cdot\vec v)))&\mbox{(associative law)}\\ \vec 0 &=(0\cdot \vec v)+\vec 0 &\mbox{(additive inverse)}\\ \vec 0 &=0\cdot \vec v &\mbox{(additive identity)} \end{align*}$$
The definition of vectors space refers to a vector space "over $\R$". This means that scalars are real numbers. In fact, vector spaces can be defined over other kinds of scalars. The restriction is that the set of scalars, together with addition and multiplication of scalars, must be a field. For example, the rational numbers, $\mathbb Q$, form a field, where two rational numbers are added or multiplied in the same way they are added and multiplied as real numbers. This means that we could define vector spaces "over $\mathbb Q$". The complex numbers, $\C$, also form a field. In fact, the last chapter in the textbook uses vector spaces over $\C.$ However, until we get to that chapter, all of our vector spaces will be vector spaces over $\R.$