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

\[\cos(2x)=\cos^2(x)-\sin^2(x)\]

or

\[\cos^2(x)+(-1)\sin^2(x)+(-1)\cos(2x)=0\]

Answer 2

Yes, going in the same way as Zarkon, these vectors are linearly dependent. So, the space spanned by \[v_1, v_2, v_3\]
has to be of dimension less than 3. It turns out, these vectors are not pairwise dependent, hence the dimension of the spanned space is 2. And one basis could be \[\{1, \cos 2x\}\]