Finite and Infinite-Dimensional Vector Spaces Examples 1

Recall from the Finite and Infinite-Dimensional Vector Spaces page that a vector space $V$ is said to be finite-dimensional if there exists a set of vectors $\{ v_1, v_2, ..., v_n \}$ in $V$ that spans $V$, that is $V = \mathrm{span} (v_1, v_2, ..., v_n)$. If the vector space $V$ cannot be spanned by a finite set of vectors from $V$, then $V$ is said to be infinite-dimensional.

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

Example 1

Show that $\wp_2 (\mathbb{R})$ is a finite-dimensional vector space by finding a set of three polynomials $\{ p_0(x), p_1(x), p_2(x) \}$ that spans $\wp_2 (\mathbb{R})$. Can $\wp_2(\mathbb{R})$ be spanned by a set of two polynomials? Can $\wp_2(\mathbb{R})$ be spanned by a set of four polynomials?

Let $p(x) = a_0 + a_1x + a_2x^2 \in \wp_2 (\mathbb{R})$. We need to find polynomials $p_0(x), p_1(x), p_2(x) \in \wp_2(\mathbb{R})$ such that for scalars $b_0, b_1, b_2 \in \mathbb{R}$ we have that $p(x)$ is a linear combination of the polynomials in $\{ p_0(x), p_1(x), p_2(x) \}$, that is:

(1)

\begin{align} \quad a_0 + a_1x + a_2x^2 = b_0p_0(x) + b_1p_1(x) + b_2p_2(x) \end{align}

Note that if we let $p_0(x) = 1$, $p_1(x) = x$, and $p_2(x) = x^2$ then:

(2)

\begin{align} \quad a_0 + a_1x + a_2x^2 = b_0 + b_1x + b_2x^2 \end{align}

Thus for any polynomial $p(x) = a_0 + a_1x + a_2x^2 \in \wp_2 (\mathbb{R})$ we have that $b_0 = a_0$, $b_1 = a_1$, and $b_2 = a_2$. Thus $\{ 1, x, x^2 \}$ spans $\wp_2 (\mathbb{R})$.

Now there does not exist any set of two polynomials $\{ q_0(x), q_1(x) \}$ where $q_0(x) = e_0 + e_1x + e_2x^2$ and $q_1(x) = f_0 + f_1x + f_2x^2$ that spans $\wp_2 (\mathbb{R})$. To show this, let $c_0, c_1 \in \mathbb{R}$. Then we'd have that:

(3)

\begin{align} \quad a_0 + a_1x + a_2x^2 = c_0q_0(x) + c_1q_1(x) \\ \quad a_0 + a_1x + a_2x^2 = c_0(e_0 + e_1x + e_2x^2) + c_1(f_0 + f_1x + f_2x^2) \\ \quad a_0 + a_1x + a_2x^2 = c_0e_0 + c_1f_0 + (c_0e_1 + c_1f_1)x + (c_0e_2 + c_1f_2)x^2 \end{align}

Thus we must have that:

(4)

\begin{align} \quad a_0 = c_0e_0 + c_1f_0 \\ \quad a_1 = c_0e_1 + c_1f_1 \\ \quad a_2 = c_0e_2 + c_1f_2 \end{align}

We can this of the constants $c_0$ and $c_1$ as unknowns to this system. Thus we have a system of three equations in two unknowns, and so for some set of scalars $a_0, a_1, a_2$ there does NOT exist a solution of scalars $c_0$ and $c_1$ for the above system. Thus any set of two polynomials does not form a spanning set of $\wp_2 (\mathbb{R})$.

Furthermore, consider the set of polynomials $\{ 1, x, x^2, 0 \}$. This set of vectors spans $\wp_2 (\mathbb{R})$ and contains four vectors, so indeed, a set of four vectors can span $\wp_2 (\mathbb{R})$.

Example 2

Show that the set of polynomials of any degree with real coefficients, $\wp (\mathbb{R})$, is infinite-dimensional.

If $\wp (\mathbb{R})$ is instead finite-dimensional, then $\wp (\mathbb{R})$ can be spanned by a finite set of polynomials, $\{ p_0(x), p_1(x), ..., p_n(x) \}$. Each of these polynomials are finite expressions, and so between each of these polynomials, there exists a polynomial with a maximum degree. Without loss of generality, assume that $\mathrm{deg} p_0 ≥ \mathrm{deg} p_j$ for $j = 1, 2, ..., n$. Let $\mathrm{deg} p_0 = m$. Then for $a_0, a_1, ..., a_m \in \mathbb{F}$ we can express this polynomial as:

(5)

\begin{align} \quad p_0(x) = a_0 + a_1x + ... + a_mx^m \end{align}

Now consider a polynomial $q(x) \in \wp (\mathbb{R})$ of degree $m + 1$. Then we have that $q(x) \not \in \mathrm{span} (p_0(x), p_1(x), ..., p_n(x))$ since the maximum degree that can be obtained by the span of these polynomials is $m$, and $q(x)$ has degree $m + 1$.

Thus $\wp (\mathbb{R})$ is an infinite-dimensional vector space.