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## Homework Statement

1.

(a) Prove that the following is a linear transformation:

##\text{T} : \mathbb k[X]_n \rightarrow \mathbb k[X]_{n+1}##

##\text{T}(a_0 + a_1X + \ldots + a_nX^n) = a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1}##

##\text{Find}## ##\text{Ker}(T)## and ##\text{Im}(T)##

(b) If ##\text{D} : \mathbb R[X]_{n+1} \rightarrow \mathbb R[X]_{n}##

##D(p) = p'##

and ##T : \mathbb R[X]_{n} \rightarrow \mathbb R[X]_{n+1}## is the transformation from part (a), prove that

##D \circ T = \text{id}## but that ##T \circ D \neq \text{id}##

## Homework Equations

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## The Attempt at a Solution

1a)

##T(a_0 + a_1X + \ldots + a_nX^n + \ldots + b_0 + b_1X + \ldots + b_nX^n) = a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1} + \ldots + b_0X + \frac{b_1}{2}X^2 + \ldots + \frac{b_n}{n+1} = T(a_0X + \frac{a_1}{2}X^2 + \ldots + \frac{a_n}{n+1}) + T(b_0X + \frac{b_1}{2}X^2 + \ldots + \frac{b_n}{n+1})##

##c \cdot (a_0 + a_1X + \ldots + a_nX^n) = ca_0 + ca_1X + \ldots + ca_nX^n##

##T(c \cdot (a_0 + a_1X + \ldots + a_nX^n) = T(ca_0 + ca_1X + \ldots + ca_nX^n) = (c \cdot a_0X) + (c \cdot \frac{a_1}{2}X^2) + \ldots + (c \cdot \frac{a_n}{n+1}) = c \cdot T(a_0 + a_1X + \ldots + a_nX^n)##

I conclude that ##T## is a linear transformation.

However, I'm not really sure how to find ##\text{Ker}(T)## and ##\text{Im}(T)## . For ##\text{Ker}(T)## for example, would it simply be something along the lines of ##T | a_n = 0 \forall a \in \mathbb k## ? Forgive me if this is a foolish question.

Part B has me confused, but mostly because I don't know how to evaluate ##D##. How does ##D(p)## relate to the form of the linear transformation I was given in part (a)?

Thank you Physics Forums for any help.

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