Question

let V be the space spanned by v1=cos^2x ,v2=sin^2x ,v3=cos2x
a)show that S={ v1,v2,v3} is not a basis for V
b) find a basis for V

Answer 1

\[cos2x=sin^2x-cos^2x\]
so v3= v2-v1 ---> S={v1,v2,v3} is linear dependent ---> it doesn't span V
b) the basis for V is the independent vectors in S, they are v1,v2 or
\[(cosx),(sinx)\]