- #1

TranscendArcu

- 285

- 0

## Homework Statement

## The Attempt at a Solution

I: P(R) → P(R) such that I(a

_{0}+a

_{1}x + ... + a

_{n}x

^{n}) = 0 + a

_{0}x + (a

_{1}x

^{2})/2 + ... + (a

_{n}x

^{n+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) = a

_{0}+ a

_{1}x. Then F(x) = a

_{0}x + (a

_{1}x

^{2})/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?