# What is meant by a linearly independent set of vectors in RR^n? Explain?

Jun 5, 2016

A vector set $\left\{{a}_{1} , {a}_{2} , \ldots , {a}_{n}\right\}$ is linearly independent, if there exists the set of scalars $\left\{{l}_{1} , {l}_{2} , \ldots , {l}_{n}\right\}$ for expressing any arbitrary vector $V$ as the linear sum $\sum {l}_{i} {a}_{i} , i = 1 , 2 , . . n$.

#### Explanation:

Examples of linear independent set of vectors are unit vectors in the directions of the axes of the frame of reference, as given below.

2-D: $\left\{i , j\right\}$. Any arbitrary vector $a = {a}_{1} i + {a}_{2} j$
3-D: $\left\{i , j , k\right\}$. Any arbitrary vector $a = {a}_{1} i + {a}_{2} j + {a}_{3} k$.

Jun 5, 2016

A set of vectors v_1,v_2,…,v_p in a vector space $V$ is said to be linearly independent $\iff$ the vector equation
${c}_{1} {v}_{1} + {c}_{2} {v}_{2} + \cdots + {c}_{p} {v}_{p} = 0$
has only the trivial solution for ${c}_{1} = {c}_{2} = \cdots = {c}_{p} = 0$.

Also, The set of vectors {v_1, . . . , v_n} ⊂ V is linearly independent $\iff$ (stands for iff) every vector v ∈ "span"{v_1, . . . , v_n} can be written uniquely as a linear combination
v = a_1v_1 + · · · + a_nv_n

Hope that helps...