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## Homework Statement

2. Let ##T: \mathbb{P}^1 \rightarrow \mathbb{R} \text{ be given by } T(p(x)) = \int_0^1 p(x)~dx## Describe ##Ker(T)## using set notation.

## Homework Equations

##p(x) \in \mathbb{P}^1~|~ p(x) = a_0 + a_1x##

## The Attempt at a Solution

##T: \mathbb{P}^1 \rightarrow \mathbb{R}## is a mapping/transformation from ##\mathbb{P}^1 \text{ to } \mathbb{R}##.

For ##p(x)## in ##\mathbb{P}^1 \rm , T(p(x))## is the image of ##p(x)## under the action of T. For each p(x) in ##\mathbb{P}^1 \rm , \int_0^1 p(x)~dx## is computed as ##T(p(x)) = Ax = \int_0^1 p(x)~dx = 0##

For p(x) to be mapped to the null space then T(p(x)) must be 0 which means that ##\int_0^1 p(x)~dx=0## which is really ##\int_0^1 a_0 + a_1x~dx = 0##.

Then by integrating we get ##a_0x + \frac 1 2 a_1 x^2 ~|_0^1##.

##a_0 + \frac 1 2 a_1 = 0##

##a_0 = -\frac 1 2 a_1##

Then

##Ker T = \{p(x) = a_0 + a_1x ~|~ a_0 = -\frac 1 2 a_1\}##