Proving Linearity of a Transformation: V=<sinx,cosx> and T: V --> V

[sana2476]

Homework Statement

Let V=<sinx,cosx> and T: V --> V be a transformation defined by T(f)=df/dx +f. Prove T is linear.

The Attempt at a Solution

T(f+g) = cosx-sinx+sinx+cosx
T(f)+T(g) = (sinx+cosx)'+sinx+cosx
= T(sinx)+T(cosx)

T(αf)=αcosx +αsinx
αT(f)= α(cosx+sinx)

Since we proved both addition and scalar, we can conclude that T is linear.

 

[Dick]

I think you are supposed to take f and g to be any linear combination of sin(x) and cos(x). I.e. f(x)=a*sin(x)+b*cos(x), g(x)=c*sin(x)+d*cos(x). So you've got the right idea, but you are only doing special cases.