Spanning Set of a Vector Space

Definition: Let $\{ v_1, v_2, ..., v_n \}$ be a set of vectors from the vector space $V$. If for every vector $v \in V$, $v \in \mathrm{span} (v_1, v_2, ..., v_n)$ then we write $V = \mathrm{span} (v_1, v_2, ... v_n)$ and we say the set of vectors $\{ v_1, v_2, ..., v_n \}$ is a Spanning Set of $V$ or Spans $V$.

From the definition of a spanning set of vectors $\{ v_1, v_2, ..., v_n \}$ of the vector space $V$, it should be clear that a set of vectors spans $V$ if any vector in $V$ can be written as a linear combination of $\{ v_1, v_2, ..., v_n \}$, that is if $v \in V$ then $v = a_1v_1 + a_2v_2 + ... + a_nv_n$ where $a_i \in \mathbb{F}$.

We will now look at some examples of spanning sets.

The Spanning Set of All Vectors from V

Consider the set of all vectors in $V$. It should be relatively obvious that $V = \mathrm{span} (V)$, because if $V = \{ v_1, v_2, ... \}$ then any $v_k \in V$ can be written as $v_k = 0v_1 + 0v_2 + ... + 0v_{k-1} + v_k + 0v_{k+1} + ...$, and so $V = \mathrm{span} (V)$. Therefore any vector space $V$ has a spanning set of vectors.

Standard Basis Vectors for Rⁿ

Consider the vector space $\mathbb{R}^n$ of $n$-component vectors. The standard basis vectors $e_1 = (1, 0, 0, ... 0)$, $e_2 = (0, 1, 0, ..., 0)$, … span $\mathbb{R}^n$, that is $\mathbb{R}^n = \mathrm{span} (e_1, e_2, ..., e_n)$ because any vector $x = (x_1, x_2, ..., x_n) \in \mathbb{R}^n$ can be written as a linear combination of the standard basis vectors:

(1)

\begin{align} \quad \quad x = (x_1, x_2, ..., x_n) = x_1(1, 0, 0, ..., 0) + x_2(0, 1, 0, ..., 0) + ... + x_n(0, 0, ..., 0, 1) = x_1e_1 + x_2e_2 + ... + x_ne_n \end{align}

A Spanning Set for P_n(F)

Recall that $\wp_n (\mathbb{F})$ represents the vector space of all polynomials whose degree is less than or equal to $n$. One such spanning set for $\wp_n (\mathbb{F})$ is the set of vectors $\{ 1, x, x^2, ..., x^n \}$, since every polynomial $p(x)$ of degree less than or equal to $n$ is written in the form:

(2)

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

Another spanning set for $\wp_n (\mathbb{F})$ is the set of vectors $\{ 1, 1 - x, 1 - x^2, ..., 1 - x^n \}$. Consider a linear combination of these vectors:

(3)

\begin{align} a_01 + a_1(1 - x) + a_2(1 - x^2) + ... + a_n(1 - x^n) \\ = a_0 + a_1 - a_1x + a_2 - a_2x + ... + a_n - a_nx^n \\ = (a_0 + a_1 + a_2 ... + a_n) + (-a_1)x + (-a_2)x^2 + ... + (-a_n)x^n \end{align}

It should be clear that any polynomial of degree less than or equal to $n$ can be written in this form. For example, the polynomial $p(x) = 3 + 2x^2 + 5x^4$ can be written in the form by letting $a_0 = 10$, $-a_2 = 2$ and letting $-a_4 = 5$ and all other $a_i = 0$

(4)

\begin{align} p(x) = (10 + (-2) + (-5)) + 2x^2 + 5x^4 \\ p(x) = 3 + 2x^2 + 5x^4 \end{align}