# Linear Dependence and Linear Independence Vectors

Linear Dependence
Let $V\left( F \right)$ be a vector space and let $S = \left\{ {{u_1},{u_2}, \ldots ,{u_n}} \right\}$ be a finite subset of $V$. Then $S$ is said to be linearly dependent if there exists scalar ${\alpha _1},{\alpha _2}, \ldots ,{\alpha _n} \in F$, not all zero, such that

Linear Independence
Let $V\left( F \right)$ be a vector space and let $S = \left\{ {{u_1},{u_2}, \ldots ,{u_n}} \right\}$ be a finite subset of $V$. Then $S$ is said to be linearly independent if,

This holds only when ${\alpha _i} = 0,\,\,\,i = 1,2,3, \ldots ,n$.

An infinite subset $S$ of $V$ is said to be linearly independent if every finite subset $S$ is linearly independent, otherwise it is linearly dependent.

Example 1: Show that the system of three vectors $\left( {1,3,2} \right)$, $\left( {1, - 7, - 8} \right)$, $\left( {2,1, - 1} \right)$ of ${V_3}\left( R \right)$ is linearly dependent.

Solution: For ${\alpha _1},{\alpha _2},{\alpha _3} \in R$.

Therefore, the given system of vectors is linearly dependent.

Example 2: Consider the vector space ${\mathbb{R}^3}\left( R \right)$ and the subset $S = \left\{ {\left( {1,0,0} \right),\,\left( {0,1,0} \right),\left( {0,0,1} \right)} \right\}$ of ${\mathbb{R}^3}$. Prove that $S$ is linearly independent.

Solution: For ${\alpha _1},{\alpha _2},{\alpha _3} \in R$.

This shows that if any linear combination of the elements of $S$ is zero then the coefficient must be zero. $S$ is linearly independent.