Let $V$ be a vector space and let $v_1,\cdots, v_n\in V$. $V$ is said to be generated by $v_1,\cdots,v_n$ if given an element $v\in V$, there exist numbers $x_1,\cdots, x_n$ such that
$$v=x_1v_1+\cdots+x_nv_n.$$
The expression $x_1v_1+\cdots+x_nv_n$ is called a linear combination of $v_1,\cdots,v_n$. The numbers $x_1,\cdots,x_n$ are called the coefficients of the linear combination.

Example. Let $E_1,\cdots,E_n$ be the standard unit vectors in $\mathbb{R}^n$. Then $E_1,\cdots,E-n$ generate $\mathbb{R}^n$.

Proof. Gievn $X=(x_1,\cdots,x_n)\in\mathbb{R}^n$,
$$X=\sum_{i=1}^nx_iE_i.$$

Proposition. The set of all linear combinations of $v_1,\cdots,v_n$ is a subspace of $V$.

Proof. Straightforward.

Example. Let $v_1$ be a non-zero element of a vector space $V$, and let $w$ be any element of $V$. The set
$$\{w+tv_1: t\in\mathbb{R}\}$$
is the line passing through $w$ in the direction of $v_1$. This line is not a subspace, however if $w=O$, it is a subspace of $V$, generated by a single vector $v_1$.

Example. Let $v_1,v_2$ be two elements of a vector space $V$. The set of all linear combinations of $v_1,v_2$
$$t_1v_1+t_2v_2: t_1,t_2\in\mathbb{R}\}$$
is a plane through the origin and it is a subspace of $V$, generated by $v_1,v_2$. The plane passing through a point $P\in V$, parallel to $v_1,v_2$ is the set
$$\{P+t_1v_1+t_2v_2: t_1,t_2\in\mathbb{R}\}.$$
However, this is not a subspace unless $P=O$.