# Let vec(v_1) = [( 2),(3)] and vec(v_1) = [( 4),(6)] what is the **span** of the vector space defined by vec(v_1) and vec(v_1)? Explain your answer in detail?

Jun 7, 2016

$\text{span} \left(\left\{{\vec{v}}_{1} , {\vec{v}}_{2}\right\}\right) = \left\{\lambda {\vec{v}}_{1} | \lambda \in F\right\}$

#### Explanation:

Typically we talk about the span of a set of vectors, rather than of an entire vector space. We will proceed, then, in examining the span of $\left\{{\vec{v}}_{1} , {\vec{v}}_{2}\right\}$ within a given vector space.

The span of a set of vectors in a vector space is the set of all finite linear combinations of those vectors. That is, given a subset $S$ of a vector space over a field $F$, we have
$\text{span} \left(S\right) = \left\{{\sum}_{i = 1}^{k} {\lambda}_{k} {s}_{k} | n \in \mathbb{N} , {s}_{i} \in S , {\lambda}_{i} \in F\right\}$

(the set of any finite sum with each term being the product of a scalar and an element of $S$)

For simplicity, we will assume that our given vector space is over some subfield $F$ of $\mathbb{C}$. Then, applying the above definition:

$\text{span} \left(\left\{{\vec{v}}_{1} , {\vec{v}}_{2}\right\}\right) = \left\{{\sum}_{i = 1}^{2} {\lambda}_{i} {v}_{i} | {\lambda}_{i} \in F\right\}$

$= \left\{{\lambda}_{1} {\vec{v}}_{1} + {\lambda}_{2} {\vec{v}}_{2} | {\lambda}_{1} , {\lambda}_{2} \in F\right\}$

But note that ${\vec{v}}_{2} = 2 {\vec{v}}_{1}$, and so, for any ${\lambda}_{1} , {\lambda}_{2} \in F$,

${\lambda}_{1} {\vec{v}}_{1} + {\lambda}_{2} {\vec{v}}_{2} = {\lambda}_{1} {\vec{v}}_{1} + {\lambda}_{2} \left(2 {\vec{v}}_{1}\right) = \left({\lambda}_{1} + 2 {\lambda}_{2}\right) {\vec{v}}_{1}$

Then, as any linear combination of ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ can be expressed as a scalar multiple of ${\vec{v}}_{1}$, and any scalar multiple of ${\vec{v}}_{1}$ can be expressed as a linear combination of ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ by setting ${\lambda}_{2} = 0$, we have

$\text{span} \left(\left\{{\vec{v}}_{1} , {\vec{v}}_{2}\right\}\right) = \left\{\lambda {\vec{v}}_{1} | \lambda \in F\right\}$