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

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

anonymous · Answer

What's a basis?

abb0t · Answer

$cos(2x) = cos^2x - sin^2x$

anonymous · Answer

:-----
A basis of a vector space V is a linearly independent spanning set.

Since it's already been given that {v1,v2,v3} spans V
You want to show that v1,v2, and v3 are linearly dependent.

abb0t · Answer

Thus the three functions are not linearly independent and do not form a minimal spanning set (basis)

abb0t · Answer

A basis is cos 2x
because $cos (2x) = 1-2sin^2(x)$
and $cos(2x) = 2cos^2(x) -1$
i.e. $sin^2(x) $ and $cos^2(x) $can be written in terms of cos(2x)

Similarly you can use $sin^2(x)$
or $cos^2(x)$ as a basis.

Any ONE of these can be used as a basis.

anonymous · Answer

Wrong, @abb0t 
It's not sufficient that you are able to express $\cos^2 x$ in terms of $\cos 2x$  it must be a LINEAR combination, which it is not.

@yantinearsita yes, you may use any TWO of these as a basis.  I'd stick to $\cos^2 x $ and $\sin^2 x$ though.

anonymous · Answer

oke, thank you so much @PeterPan and @abb0t for helping me to solve my homework

anonymous · Answer

Remember, though, one of them is not enough for a basis...

anonymous · Answer

ok, i get it