# Linear independence

by sncum
Tags: independence, linear
 P: 14 Any one plz tell me about the term linear independence?and when we say that the function is linear independent
 Sci Advisor P: 6,040 You need to supply context. Usually linear independence refers to a set of vectors or a set of functions. These things are called linearly dependent if some linear combination = 0. If not then they are linearly independent.
 P: 14 [A][/A]=[c][/1]x+[c][/2]y[j][/j]+[c][/3]z[k][/k] when we say it is linearly independent and also my friend argue with me that orthonormality implies linear independence but i was not satisfied plz help
P: 6,040
Linear independence

 Quote by sncum [A][/A]=[c][/1]x+[c][/2]y[j][/j]+[c][/3]z[k][/k] when we say it is linearly independent and also my friend argue with me that orthonormality implies linear independence but i was not satisfied plz help
I don't understand your first line.

However your friend is correct, if two vectors are orthogonal (unless one of the is 0) they are linearly independent. Note that the converse is not true.
Emeritus
$$\sum_{k=1}^n a_k x_k=0\quad \Rightarrow\quad a_1=\dots=a_n=0.$$
$$0=\langle e_i,0\rangle=\langle e_i,\sum_k a_k e_k\rangle=\sum_k a_k\langle e_i,e_k\rangle=\sum_k a_k\delta_{ik}=a_i.$$