- #1
bornofflame
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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\}##