Integration as a Linear Transformation

[TranscendArcu]

Homework Statement

The Attempt at a Solution

I: P(R) → P(R) such that I(a₀+a₁x + ... + a_nxⁿ) = 0 + a₀x + (a₁x²)/2 + ... + (a_n 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?

 

[Fredrik]

Let U,V be vector spaces. A function T:U→V is said to be linear if T(ax+by)=aTx+bTy for all scalars a,b and all vectors x,y.

If U=V=ℝ, then the T defined by T(x)=ax+b is clearly not linear, unless b=0. (When U=V=ℝ, both the "scalars" and the "vectors" are members of ℝ). You're working with a different space, and I didn't even read your problem carefully, but my point is just that you need to use the definition of "linear" above, instead of your intuition about what a "line" is. That's why I'm mentioning that the only functions from ℝ to ℝ that are linear are the ones whose graphs are straight lines through the origin.

 

[TranscendArcu]

And doesn't integration satisfy your definition of linear?

∫(f(x) + g(x)) dx = ∫f(x)dx + ∫g(x)dx, for two functions, f,g, in P(R)
∫r(f(x))dx = r∫f(x)dx, for some r in R

Is that not true? And isn't it true that linear transformations must map the zero vector of the domain to the zero vector of the codomain? Maybe I'm misunderstanding.

 

[HallsofIvy]

Note that the statement of the problem specifically said "Let F(x) be the polynomial with F(0)= 0 such that F'= f".

If that condition "with F(0)= 0" were not there, this "theorem" would not be true.

 

[TranscendArcu]

Does that indicate that integration without the condition "F(0) = 0" is not a linear transformation?

 

[Fredrik]

Does that indicate that integration without the condition "F(0) = 0" is not a linear transformation?

Perhaps you can tell us. If f,g are arbitrary polynomials, a,b are arbitrary real numbers, and I denotes the map $f\mapsto F$, what is I(af+bg)?