Making a Set a Basis: Is V Spanned?

[torquerotates]

Ok, so I have the set S={cosx^2, sinx^2, cos(2x)} that spans V.

So obviously this is not a basis. Because the third is a linear combination of the first; ie. cosx^2-sinx^2=cos(2x). But if I were to take the first 2 vectors, would they span V? That is does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

 

[George Jones]

Ok, so I have the set S={cosx^2, sinx^2, cos(2x)} that spans V.

So obviously this is not a basis. Because the third is a linear combination of the first; ie. cosx^2-sinx^2=cos(2x). But if I were to take the first 2 vectors, would they span V? That is does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

Do you mean: Are the first two vectors linearly independent?

Use $\cos^2x = 1 - \sin^2x$.

 

[gel]

does the equation a(cosx^2)+b(sinx^2)=0 imply a=0& b=0?

try evaluating at x=0 and x=$\pi/2$