Finite and Infinite-Dimensional Vector Spaces

Definition: A vector space $V$ which is spanned by a finite set of vectors $\{ x_1, x_2, ..., x_m \}$ is said to be a Finite-Dimensional Vector Space. If $V$ cannot be spanned by a finite set of vectors then $V$ is said to be an Infinite-Dimensional Vector Space.

We will now look at some examples of finite and infinite-dimensional vector spaces.

Standard Basis Vectors for Rⁿ

Consider the vector space $\mathbb{R}^n$, and the set of vectors in $\mathbb{R}^n$, and consider the vectors $e_1 = (1, 0, 0, ... 0)$, $e_2 = (0, 1, 0, ..., 0)$, …, $e_n = (0, 0, ..., 1)$. These vectors are known as standard basis vectors and $\mathbb{R}^n = \mathrm{span} \{ e_1, e_2, ..., e_n \}$ since any vector $x \in \mathbb{R}^n$ can be written as a linear combination of these vectors:

(1)

\begin{equation} x = k_1e_1 + k_2e_2 + ... + k_ne_n = k_1(1,0,0,...,0) + k_2(0,1,0,...,0) + ... + k_n(0,0,0,...,0) \end{equation}

The Set of Monomials for P_n(F)

Consider the vector space $\wp _n (\mathbb{F})$, which was the set of polynomials $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$ whose degree $\deg p ≤ n$ and where $a_0, a_2, ..., a_n \in \mathbb{F}$.

Now consider the set of vectors that are simple monomials, that is the set $\{ 1, x, x^2, ..., x^n \}$. We note that $\wp _n(\mathbb{F}) = \mathrm{span} \{ 1, x, x^2, ..., x^n \}$ since any vector $p(x) \in \wp _n (\mathbb{F})$ can be written as a linear combination of these vectors, that is:

(2)

\begin{equation} p(x) = a_01 + a_1x + a_2x^2 + ... + a_nx^n \end{equation}

The Set of Monomials for P(F)

Recall that the vector space $\wp (\mathbb{F})$ is the set of all polynomials with arbitrary degree, that is $p(x) = a_0 + a_1x + a_2x^2 + ...$. We note that vector space is infinite-dimensional.

Suppose not, and suppose there exists a finite set of vectors such that any vector $p(x) \in \wp (\mathbb{F})$ can be written as a linear combination of these vectors in this finite set. Since this set is finite, there contains some vector with a largest degree, call it $m$. But then any vector $p(x)$ such that $\deg p > m$ cannot be represented as a linear combination of the vectors in this finite set, so our assumption that $\wp (\mathbb{F})$ is finite-dimensional was false.