Question

Convert \(P_2\) inti an inner product space by writing (x,y) =\(\int_0^1 x(t)\bar y(t) dt\) whenever x, y are in \(P_2\) and find a complete orthonormal set in that space
Please, help

Answer 1

since x, y in P2, they have the same basis, right?

Answer 2

let say, the basis is {1,t,t^2}

Answer 3

unless convert it to dual basis (1,0,0), (0,1,0), (0,0,1) I don't know how to construct the orthonomal basis set

Answer 4

i am probably way off base, but isn't \(\{1,t,t^2\}\) a complete orthonormal basis?

Answer 5

the norm of each of them is 1 and it is certainly complete

Answer 6

but i would not be $12 that it is correct, but it sure seems correct

Answer 7

*bet

Answer 8

how can you know it is perpendicular to each other or not? they are l.i

Answer 9

As I said above, unless convert to standard basis, I have no way to prove those vectors orthogonal, then orthonomal set

Answer 10

oh crap
i guess you have to do that gramm schmidt business

Answer 11

What are x and y?

Answer 12

polynomials in P2

Answer 13

i take it they are polynomials of degree 2 but i could be wrong

Answer 14

So \[
\sum_{n=0}^2a_nx^n
\]

Answer 15

But what would the conjugate be?

Answer 16

I think we proceed on complex space first and if argue on real then.

Answer 17

So then the coefficients and \(t\) are complex numbers?

Answer 18

My prof said \(\bar y(t)\) means conjugate coefficients of y

Answer 19

oh maybe it is not that bad
put \(u_1=1\) then \(u_2=t-\int_0^11\times tdt=xt-1\)

Answer 20

typo there
i meant \(u_2=t-\int_0^11\times tdt=t-1\)

Answer 21

you also have to divide by the norm or each vector, but the norm is one in each case so not an issue here

Answer 22

no i am screwing this up

Answer 23

What?\[
\int_0^1(1\times t)\;dt = \left[\frac{t^2}{2}\right]_0^1 = \frac 12
\]

Answer 24

yeah i am messing this up totally
but now what we have that, we have \(u_2=t-\frac{1}{2}\)

Answer 25

Well, what I can say is we need \(\|x\| = (x_1,x_1) = 1^2=1\) and we need \((x_1,x_2)=0\).

Answer 26

then you need the norm of \(t-\frac{1}{2}\) and divide by that

Answer 27

Maybe try \(a\), \(bt\) and \(ct^2\), where \(a,b,c\in \mathbb C\) and try to solve for those coefficients?

Answer 28

i was trying gramm schmidt
are you supposed to do something else?

Answer 29

One question: What is Rieze Representation Theorem? does it work for this problem?

Answer 30

i get if \(w_1=1\) then \(w_2=12t-6\)

Answer 31

no idea
but at least i have produced two that are orthogonal
this gramm schmidt actually seems to work

Answer 32

I am getting \((w_2,w_2) = 12\)... or am I suppose to do something weird to get \(\overline{w_2}\)?

Answer 33

i am still messing up
let me go slower

Answer 34

I think I know what you're trying to do now.

Answer 35

You are getting a projection using the provided inner product. But how would you ever get any complex coefficients doing this? Or does not not matter?

Answer 36

i forgot to take the damned square root!

Answer 37

ok finally i think i have the first two
\(1\) and \(2\sqrt3(t-\frac{1}{2})\)

Answer 38

i will probably make four or five mistakes before i find the third one

Answer 39

i can walk you through what i did if you like but if you are supposed to use something other than gramm schmidt i don't know it

Answer 40

I understand what you are doing, my question is you are trying to find an orthonomal set on the given basis, how it relates to the (x,y) = int.....? how the dual basis y which supposed to be orthonomal dual of x is defined?

Answer 41

Using Gramm Schmidt to orthonomal the given basis is not hard, (just be patient), how to put it in use is what I want to know. :)

Answer 42

maybe i don't understand the question
i thought you were looking for an orthonormal basis

Answer 43

You can use orthonormal basis to project one vector subspace onto another vector subspace.

Projections are good for finding the solution with the least error, when the exact solutions are in another subspace.

Answer 44

I have a note from my prof, however, I don't understand what he was talking about. Let me copy it
V = P_n
define (x,y)= \(\int_a^b x(t)\bar y(t) dt\) an inner product on P_n
Let f in P'_n given by [x,f]= \(x(t_0)\) in Complex
By Riesz Representation Theorem, there exists unique P in P_n such that
\(\int_a^b x(t)\bar P(t) dt = x(t_0)\) for every x in P_n
P depends on n, a, b, \(t_0\)
end note.