# Linear Transformation

## 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.