Question

Determine if the following is a linear transformation. If it is, find the standard matrix of transformation.
T:P1-->P2 such that T(p(t)) = integral of p(t)dt from 0 to x. (I don't know how to make an integral sign on this. sorry!)

Answer 1

\[\int\limits_{0}^{x} p(t) dt\]

Answer 2

do u know the two basic properties of integrals?

Answer 3

In order for it to be a linear transformation T(V1+V2)= T(V1) + T(V2) and T(kV1)=k T(V1).
Right?

Answer 4

\[ \large \int(A+B)=\int A+\int B \]
and
\[ \large \int(\alpha K)=\alpha\int K \]

Answer 5

Oh, those rules. Yes I am familiar with them. I misinterpreted what you were asking for.

Answer 6

so for instance
\[ \large T(p_1(t)+p_2(t))=\int_0^x[p_1(t)+p_2(t)]\,dt \]
\[ \large =\int_0^xp_1(t)\,dt+\int_0^xp_2(t)\,dt=T(p_1(t))+T(p_2(t)) \]

Answer 7

Alright, I follow you so far. How would go about finding out if it was a linear transformation?

Answer 8

one step missing.
\[ \large T(\alpha p_1(t))= \]

Answer 9

Ok, and those steps illustrate that it is a linear transformation?

Answer 10

they don't "illustrate" they demonstrate that it IS a linear transformation.

Answer 11

complete the second step. to see if u got it.

Answer 12

\[\alpha\] is a constant so it can be moved out side leaving you with
\[\alpha \]\[P _{1}(t)\] which demonstrates that it is a linear transformation.

Answer 13

\[ \large T(\alpha p_1(t))=\int_0^x[\alpha p_1(t)]\,dt=
\alpha\int_0^xp_1(t)\,dt=\alpha\cdot T(p_1(t)) \]

Answer 14

now. u have demonstrated that T is in fact a linear transformation.

Answer 15

ok, that part of the problem makes it so much more sense now! Thank you!
to find the standard matrix of the transformation, would you simply use the basis {1, t, \[t^{2}\] }?

Answer 16

yes. i guess that is what it means.

Answer 17

ok. thank you so much for your help!

Answer 18

u have the following
\[ \large P_1=\langle1,x\rangle \]
and
\[ \large P_2=\langle1,x,x^2\rangle \]

Answer 19

right?

Answer 20

yes

Answer 21

sorry. a typo (considering the wording of your problem)
\[ \large P_1=\langle1,t\rangle \]
and
\[ \large P_2=\langle1,t,t^2\rangle \]

Answer 22

now compute
\[ \large T(1)= \]
and
\[ \large T(t)= \]

Answer 23

I feel like I'm making an error (big surprise)
I get T(1)=x and T(t)=\[x^{2}\]

Answer 24

check the second one.

Answer 25

T(t)=\[\frac{ x^{2} }{ 2 }\]?

Answer 26

great.

Answer 27

so now you have:
\[ \large T(1)=x=\color{red}{0}\cdot1+\color{red}{1}\cdot x+\color{red}{0}\cdot x^2 \]
and
\[ \large T(t)=\frac{x^2}{2}=\color{red}{0}\cdot1+\color{red}{0}\cdot x
+\color{red}{\frac{1}{2}}\cdot x^2 \]
right?

Answer 28

can u write the matrix from this?

Answer 29

\[\left[\begin{matrix}0 & 0 \\ 1 & 0 \\ 0 & 1/2\end{matrix}\right]\]?

Answer 30

yes. good work.

Answer 31

Just to clarify, that is the standard matrix of the transformation?

Answer 32

yes. because u used the standard basis of each space.

Answer 33

I understand this problem so much better now. Thank you so much for your help. I really appreciate it!

Answer 34

u r very welcome