prove that the members of an orthogonal set of functions are linearly independent.?

Answer 1

let {V₁, V₂, V₃, V₄,..., Vn} be orthogonal

then for some Vi in the set, and consider

c₁V₁ + c₂V₂ + c₃V₃ + ... + ciVi + ...cnVn = 0

get its dot product with Vi
c₁V₁•Vi + c₂V₂•Vi + c₃V₃•Vi + ... + ciVi•Vi + ...cnVn•Vi
=0 + 0 + ...+ ci||Vi||² + ... + 0 = 0
Since ||Vi|| ≠ 0 , then ci = 0 for all i.

Then the set is linearly independent. d:

Answer 2

Come on, man; orthogonal set says everything! That is why they are linearly independent: because they are orthogonal.

Try to apply the definitional relation and you will see that these functions are LI.

(But I guess you are too lazy; OMFG!)

Answer 3

This is almost definition. If they were NOT linearly independent, you could form any member from any other members of the set. Since they are orthagonal, this cant be done (with the simple axis in the x,y, and z direction), you cannot form a vector ax+by+cz with only x and y vectors.