1. The problem statement, all variables and given/known data 3. The attempt at a solution I: P(R) → P(R) such that I(a0+a1x + ... + anxn) = 0 + a0x + (a1x2)/2 + ... + (an xn+1)/(n+1) Clearly this is just integration such that c = 0. It is easily shown that integration is a linear transformation, so I conclude that I is a linear transformation. However, in calculus we do not usually require that c = 0. Yet integration, if it is a linear transformation, must map the zero vector of the domain to the zero vector of the codomain. Let f(x) = a0 + a1x. Then F(x) = a0x + (a1x2)/2 + c. A polynomial is zero iff all of its coefficients are zero. If we decide that c is not necessarily zero, doesn't this iff fail? That is, is integration a linear transformation only when we stipulate that c=0?