Some properties of span of a vector space

In this post, we consider “span” of a vector space $V$. As we know, definition of span is:

Definition 1 Let ${\{v_1, v_2, \dots, v_n\}}$ be a finite set of vectors in a vector space ${V}$. The subset of ${V}$ spanned by ${\{v_1, v_2, \dots, v_n\}}$ is the set of all linear combinations of ${v_1, v_2, \dots, v_n}$. This set is called the span of ${\{v_1, v_2, \dots, v_n\}}$ and is denoted

$\displaystyle \text{span}\{v_1, v_2, \dots, v_n\}.$

For example, you can see

Example 1 Let ${\{v_1=(1,0), v_2=(1,1)\}}$, we have ${span\{v_1, v_2\}=\{x(1,0) + y(1,1)| x, y \in \mathbb{R}\}}$. Of course, ${span\{v_1, v_2\} \subset \mathbb{R}^2}$. It is easy to see that ${\mathbb{R}^2 \subset span\{v_1, v_2\}}$ because ${v = (x',y') \in \mathbb{R}^2 \Rightarrow (x',y') = x(1,0)+y(1,1) \Leftrightarrow \begin{cases} x'&= x+ y\\ y'& = y \end{cases} }$. That system has always solution ${x, y \in \mathbb{R}}$. This implies ${span\{v_1, v_2\} = \mathbb{R}^2}$.

We have the following result

Theorem 2 The span of any finite set ${\{v_1, v_2, \dots, v_n\}}$ of vectors in a vector space ${V}$ is a subspace of ${V}$.

Some properties:

1. Suppose ${W}$ is a subspace of a vector space ${V}$. Prove that if ${v_1, v_2, \dots, v_n \in W}$, then ${span\{v_1, \dots, v_n\} \subseteq W}$.
2. Suppose ${v_1, \dots, v_m}$ and ${w_1, \dots, w_n}$ are vectors in a vector space satisfy:
if ${w_1, \dots, w_n \in span\{v_1, \dots, v_m\}}$ and ${v_1, \dots, v_m \in span\{w_1, \dots, w_n\}}$ then ${span\{v_1, \dots, v_m\} = span\{w_1, \dots, w_n\}}$.

References:

R. Messer, Linear algebra gateway to mathematics, Harper Collins College Publishers

D. Lay, Linear algebra and its applications, Addison-Wesley, 2012.

(cont.)