Recall our definition of a vector space, which is an additive abelian group with a field action.
A
vector space over a field \(\mathbb{F}\) is a set \(V\) with two operations:
- Addition: \(+:V \times V \rightarrow V\), and
- Scalar multiplication: \(\cdot : \mathbb{F} \times V \rightarrow V \)
such that the following axioms hold for all \(u,v,w \in V\) and all \(a,b \in \mathbb{F}\).
- Commutativity. \(u+v = v+u\)
- Associativity. \( (u+v)+w = u+(v+w) \) and \( (ab)v = a(b v) \)
- Additive Identity. Exists \(0 \in V\) such that \(v + 0 =v \)
- Additive Inverse. For all \(v \in V\) there exists \(z \in V\) such that \(v+z=0\)
- Multiplicative Identity. \(1v=v\)
- Distributive Properties. \(a (u + v) = au + av\) and \(a+b)u = au + bu\)
The first four axioms (excluding the one referring to the field) say that \(V,+\) is an additive abelian group, while the rest describe the field action.
Terminology and asides
Note that “a vector space over \(\mathbb{R}\)” and “a real vector space” mean the same thing. For \(V\) a vector space, we call \(v \in V\) a vector, element, or point.
We saw how thinking geometrically can help:
-
it reminds you that a vector is NOT a list of numbers until you choose some reference vectors, and
-
some properties of vectors and vector spaces become intuitively obvious.
But, be warned: 2D and 3D intuition can lead you astray when considering high-dimensional, infinite dimensional, complex vector spaces, and other hard-to-visualize spaces. Even 4D is hard.
During this course you will start to acquire a mixed intuition: 2D and 3D geometry plus some warning buzzers around special properties of these low-dimensional real vector spaces.
Examples
A set with a single element (the additive identity 0) is a vector space over any field. What is vector addition and scalar multiplication?
The \(\mathbb{R}\)-valued functions on the interval [0,1] form a real vector space. What is vector addition and scalar multiplication?
The linear transformations between two finite-dimensional vector spaces themselves form a vector space.
In these examples I want to emphasize that
- a vector (that is, an element of a vector space) is NOT (necessarily) a list of numbers,
- a vector space is not the set of all (a,b,c), and
- a linear transformation is not always a matrix!
That said, let’s consider an example of that sort.
\(\mathbb{F}^n\) denotes the vector space of ordered n-tuples of field elements, with field addition and broadcast scalar multiplication. If \(n=2\), addition is
$$
\begin{pmatrix}
x \\
y
\end{pmatrix}
+
\begin{pmatrix}
z \\
w
\end{pmatrix}
=
\begin{pmatrix}
x+z \\
y+w
\end{pmatrix}
$$
and scalar multiplication, with \(\alpha \in \mathbb{F}\), is
$$
\alpha
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
\alpha x \\
\alpha y
\end{pmatrix}
$$
Here is an example of a vector space which has countably infinite dimension.
A
univariate polynomial is a function from \(\mathbb{F}\) to \(\mathbb{F}\) given by
$$
p(x) = a_0 + a_1 x + a_2 x + \cdots + a_m x^m
$$
where \(m\) is the
degree of the polynomial, for some \(a_0, a_1, \dots, a_m \in \mathbb{F}\).
Suppose \(p\) and \(q\) are polynomials, e.g. \(p(x)=3x^2+1\) and \(q(x) = x^4 + 2x^3 +5x^2\).
We define the polynomial \(p+q\) by
$$
(p+q)(x) = p(x)+q(x)
$$
so, with our example polynomials,
$$
(p+q)(x) = x^4 + 2x^3 + 8x^2 +1.
$$
Here we are defining
addition of polynomials in terms of
addition of real numbers. Similarly, scalar multiplication is defined for \(\alpha \in \mathbb{F}\) by
$$
(ap)(x) = ap(x)
$$
Again all we need to do to make definitions here is to change the order of parenthesis to define scalar multiplication of polynomials (left hand side) in terms of scalar multiplication of real numbers (right hand side). For example, with our example \(p\),
$$
7p(x) = 21x^2 +7.
$$
The additive identity element in the vector space of univariate polynomials is the
zero polynomial defined by \(a_0 = 0, a_1 = 0, \dots \) and denoted by 0.
Zeros in a field, and zero vectors (such as the zero polynomial) are generally denoted by the same symbol, 0, although they live in different spaces. It can be helpful to distinguish them by a subscript, e.g. in the identity \(0_{\mathbb{F}}v =0_{V}\). This identity means that multiplying any vector \(v\) by the zero of the field \(0_{\mathbb{F}}\) yields the zero \(0_V\) of the vector space.