Let $V$ be a vector space. $v_1,\cdots,v_n\in V$ are said to be linearly dependent if there exist numbers $a_1,\cdots,a_n$ not all equal to $0$ such that
$$a_1v_1+\cdots+a_n=O.$$
If there do not exist such numbers, then we say $v_1,\cdots,v_n$ are linearly independent. That is, $v_1,\cdots,v_n$ are linearly independent if whenever $a_1v_1+\cdots+a_nv_n=O$, $a_1=\cdots=a_n=0$.

Example. In $\mathbb{R}^n$, the standard unit vectors $E_1,\cdots,E_n$ are linearly independent.

Example. The vectors $(1,1)$ and $(-3,2)$ are linearly indepdent in $\mathbb{R}^2$.

A set of vectors $\{v_1,\cdots,v_n\}\subset V$ is said to be a basis of $V$ if $v_1,\cdots,v_n$ generate $V$ and that they are linearly independent.

Example. The vectors $E_1,\cdots,E_n$ form a basis of $\mathbb{R}^n$.

Example. The vectors $(1,1)$ and $(-1,2)$ form a basis of $\mathbb{R}^2$.

In general for $\mathbb{R}$, the following theorem holds.

Theorem. Let $(a,b)$ and $(c,d)$ be two vectors in $\mathbb{R}^2$.

(i) They are linearly depedendent if and only if $ad-bc=0$.

(ii) If they are  indepdendent, they form a basis of $\mathbb{R}^2$.

Proof. Exercise

Let $V$ be a vector space and let $\{v_1,\cdots,v_n\}$ be a basis of $V$. If $v\in V$ is written as a linear combination
$$v=x_1v_1+\cdots+x_nv_n,$$
$(x_1,\cdots,x_n)$ is called the coordinates of $v$ with respect to the basis $\{v_1,\cdots,v_n\}$. For each $i=1,\cdots,n$, $x_i$ is called the i-th coordinate. The following theorem says that there can be only one set of coordinates for a given vector.

Theorem. Let $V$ be a vector space. Let $v_1,\cdots,v_n$ be linearly independent elements of $V$. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are numbers such that
$$x_1v_1+\cdots+x_nv_n=y_1v_1+\cdots+y_nv_n,$$
then $x_i=y_i$ for all $i=1,\cdots,n$.

Proof. It is strightforward from the definition of linearly independent vectors.

Example. Find the coordinates of $(1,0)$ with respect to the two vectors $(1,1)$ and $(-1,2)$.

Example. The two functions $e^t$ and $e^{2t}$ are linearly independent.

Proof. Exercise.

Theorem. Let $v,w$ be two vectors of a vector space $V$. They are linearly dependent if and only if one of them is a scalr multiple of the other, i.e there is a number $c$ such that $v=cw$ or $w=cv$.

Proof. Exercise.

If one basis of a vector space $V$ has $n$ elements and another basis has $m$ elements, then $n=m$. The number of elements in any basis of a vector space $V$ is called the dimension of $V$ and is denoted by $\dim V$.