Question

3. Define a transformation from P2 to R3 by:

\[p(x) -> ( \int\limits_{0}^{1} p(x) d(x) , \int\limits_{0}^{2} p(x) d(x) , \int\limits_{0}^{3} p(x) d(x) )\]

(a) Is this a homomorphism?
(b) Is it onto?
(c) Is it one-to-one?
(d) Is it an isomorphism?
(e) Find a matrix for the transformation between the representation of the standard basis of P2 and the standard basis of R3.

Answer 1

2 spaces

Answer 2

Basically transforming a vector with two entries to a vector with 3 entries right?

Answer 3

This is a linear transformation right?

Answer 4

yea

Answer 5

do you have an idea?!

Answer 6

help @eliassaab

Answer 7

It is linear since integration is linear.

Answer 8

so what 's the answer for these questions?

Answer 9

Well the least you can do is write down the definition for each one. If no one answers then just ask your prof.

Answer 10

It is onto
since if you take \[

(u,v,w)\in R^3\\
p(x)=3 u - (3 v)/2 + w/3 + (-5 u + 4 v - w) x + 1/2 (3 u - 3 v + w) x^2
\]
then the image of p(x) is (u,v,w)

Answer 11

It is one to one since both spaces are of dimension 3 and there is an onto map from
one to the othr

Answer 12

I thought you couldnt have both.

Answer 13

Thank you prof. @eliassaab

Answer 14

The image of the polynomial 1 is
{1,2,3}

The image of the polynomial x is\[

\left\{\frac{1}{2},2,\frac{9}{
2}\right\}
\]

The image of the polynomial x^2 is\[

\left\{\frac{1}{3},\frac{8}{3}
,9\right\}
\]

Answer 15

The matrix of the transformation is
\[
\left(
\begin{array}{ccc}
1 & \frac{1}{2} & \frac{1}{3}
\\
2 & 2 & \frac{8}{3} \\
3 & \frac{9}{2} & 9 \\
\end{array}
\right)
\]