Inner Product Space Isomorphism Help Needed!

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[balmy linden]

What is p(0) and p(1) ?

[regal kiln]

map basis to basis

[inner estuary]

How so?

The basis for the inner product space is {1, x}

and the basis for R^2 is {(1,0) , (0,1)}

[balmy linden]

don't you need to verify that the map preserves orthogonality

[regal kiln]

there are many bases on r2 but that one is fine too

[inner estuary]

That is just the standard basis for R^2

How do I map?

[regal kiln]

you have to preserve inner product

[regal kiln]

try something, you have two elements to choose between

[inner estuary]

I'm not sure how to show this

[inner estuary]

Or this

[regal kiln]

definitions.. 😖

[balmy linden]

I never studied projective geometry that's why I asked what p(1) and p(0) were, but if I knew better I'd tell you to stare long enough at the defined inner product

There's probably a reason for it being defined that way

[regal kiln]

what does it mean for a map to "preserve inner product"

[balmy linden]

in particular, you know the inner product in R² quite well

so you can try reverse-engineering the whole thing to find a map that fits your requirements

what do you have to map to (1, 0) and what do you have to map to (0, 1)?

[inner estuary]

Do I need to prove it's linear, one to one and onto?

[regal kiln]

no

your only concern whether it preserves the inner product

[balmy linden]

if f is your isomorphism, a desideratum is that <f(x), f(y)> = <x, y>

[inner estuary]

I'm not sure what preserves inner product actually means

[regal kiln]

he just said what it means

[inner estuary]

Um, can I get a simple example of that?

[balmy linden]

but up to a scaling, it doesn't matter too much (you can rescale afterwards). What really matters to you right now is that right angles are preserved

[regal kiln]

you got the definition, now start trying things

[inner estuary]

Is that the only thing needed to find an isomorphism?

[regal kiln]

map the basis somehow and see if the inner product is preserved

[regal kiln]

if you wanna do it by definition then thats fine too

find two mutually inverse inner product isomorphisms

you have to know what this means..

[balmy linden]

As for the technical details I am afraid I cannot help much without better knowledge of projective geometry, so I'll just step out

[inner estuary]

Ok I'll start trying stuff and get back to u

[regal kiln]

are you also required to verify the thing on P1 is an inner product?

[inner estuary]

Is that part of finding an inner product isomorphism?

[obtuse kayak]

Generally if $(E,\langle .|. \rangle)$ is a Euclidean space (of dimension n) and $\mathcal{B}=(e_i){1 \le i \le n}$ is an orthonormal basis of $E$ then the map $\phi{\mathcal{B}}: x=\sum_{i=1}^{n} x_i e_i \mapsto (x_i){1 \le i \le n}$ is an isomorphism from $E$ to $\bR^{n}$ and if $\bR^{n}$ is given it’s usual inner product $\langle .|. \rangle_1$ then first prove that $\phi{\mathcal{B}}$ preserves inner product so $\langle x|y \rangle=\langle \phi_{\mathcal{B}}(x)| \phi_{\mathcal{B}}(y) \rangle_1$ for any vector x and y in E

Once you have proved this, you can try finding an orthonormal basis of P1(R) for the inner product then you have an isomorphism that satisfies the conditions

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[obtuse kayak]

That is assuming R^2 is equipped with its usual inner product

[inner estuary]

Am I assuming that for this question?

[obtuse kayak]

If the inner product is not specified for R^2 then I would assume so yes

[inner estuary]

that given inner product is in P_1(R)

[obtuse kayak]

Yeah so consider that R^2 is given its usual inner product and P_1(R) is given the inner product shown in the exercise

[inner estuary]

ok

and then?

[obtuse kayak]

Do this

Find an orthonormal basis of P_1(R) for the innner product then you can construct an isomorphism

[inner estuary]

I don't have to do this

[inner estuary]

Ok done.

[obtuse kayak]

What basis did you find ?

Seems good

[inner estuary]

Not sure how to proceed now

[obtuse kayak]

Now you can construct an isomorphism that preserves inner product

The isomorphism in question is the application that for a polynomial P=aX+b maps its coordinates in the orthonormal basis that you found

[inner estuary]

Are we mapping the orthonormal basis to the standard basis of R^2?

[obtuse kayak]

if (e_1,e_2) is your basis you are considering the map P=a_1e_1+a_2e_2–>(a_1,a_2)

so for a fixed polynomial P=aX+b you need to find a_1 and a_2

And this can be done easily because (e_1,e_2) is orthonormal (the coefficients in the basis can be expressed via the inner product)

[obtuse kayak]

Here for any $i$, $x_i=\langle x|e_i \rangle$

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[obtuse kayak]

So the map in question is basically $x \mapsto (\langle x|e_i \rangle )_{1 \le i \le n}$

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[obtuse kayak]

(Because B is orthonormal)

[inner estuary]

So how do we write out L?

[obtuse kayak]

Take any polynomial P=aX+b what is $\langle P | e_1 \rangle$?

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[obtuse kayak]

e_1 is the first element in your orthonormal basis

And do the same with e_2, e_2 being the second element in your basis

then you can take $L:P \mapsto (\langle P |e_1 \rangle, \langle P|e_2 \rangle )$

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[inner estuary]

What does the | mean?

[regal kiln]

notation

[obtuse kayak]

Here $\langle .|. \rangle$ denotes the inner product

[untold prairie]

😑 ɿototoЯ | Rototor 😑

[inner estuary]

is it the same as comma?

[regal kiln]

yes

[inner estuary]

same as this?

[regal kiln]

yes

[obtuse kayak]

Oh yes, I’m just used to this notation, sorry for the confusion

[inner estuary]

So this is our inner product isomorphism?

Like this would be the answer to the question?

but we need to show that it preserves inner products?

[obtuse kayak]

If (e_1,e_2) is the orthonormal basis you found then yes it would be the answer to the question

Try proving that it preserves inner product

[inner estuary]

How come we didn't mention anything about the basis of R^2 tho?

Where did we map out orthonormal basis to R^2?

[regal kiln]

dude..calculate something

he has told you more than enough to solve the problem

[regal kiln]

show that the thing you map to IS a basis in R2

and he did say where to map it

[inner estuary]

So we r doing this?

[balmy linden]

It is indeed exactly what I told you at the beginning of this post

The preservation of the inner product I mean

[inner estuary]

Hmm ok

I'm still very confused for what exactly I need to calculate

I calculated the orthogonal basis for the inner product

[balmy linden]

I will ask to clarify

What is the first vector space? P1(R)

[inner estuary]

Polynomials

[balmy linden]

Ok this whole time I thought it was a projective line

[balmy linden]

You need to come up with an isomorphism

And prove that it is an isomorphism that is inner product preserving

[inner estuary]

Yes but how is what I'm still lost about 😭

[balmy linden]

Which part

It would help a lot to pinpoint what part is causing you confusion

We can hardly help you if you say "all of it"

[inner estuary]

Well firstly what is the final form of the answer supposed to look like?

What is the form of "find an inner product isomorphism"

[balmy linden]

An isomorphism

That is, a linear map that is bijective

you want an isomorphism that verifies the property indicated above (inner product preservation)

[inner estuary]

Where does the basis for R^2 come into play?

[balmy linden]

You should have seen in class that between finite dimensional vector spaces, a linear map is characterized by the image of a basis

So it suffices to know where to map basis vectors of P1(R)

Given the desideratum that you must also preserve orthogonality

This is the reason why it suffices to map an orthogonal basis of P1(R) to an orthogonal basis of R²

Now the most obvious orthogonal basis of R² is the standard basis (assuming you're using the standard dot product)

[inner estuary]

So is my isomorphism then just L(P) = (a_1 a_2) ?

Also, since e1 and e2 are orthonormal is is true that, ⟨e1, e1⟩ = 1, ⟨e2, e2⟩ = 1, and ⟨e1, e2⟩ = 0?

So is this my isomorphism because of the following:

⟨P, e1⟩ and ⟨P, e2⟩ are the inner products of P with the basis elements e1 and e2 in P1_R
​

This map L takes any polynomial P(x) = ax+b in P_1(R) and maps it to a vector in R^2 whose coordinates are determined by the inner products with the orthonormal basis {e1, e2} of P_1(R)?

and since {e1, e2} is orthonormal, L will preserve the inner product, making it an isomorphism of inner product spaces?

Also, in R^2, isn't the standard inner product of two vectors (a,b) ⋅ (c,d) = ac + bd ?

[balmy linden]

This looks alright

[balmy linden]

Insufficient justification imo

You need to prove the following things

1. is L linear?
2. is L bijective?
3. is L inner product preserving?

And here you can do it methodically

[inner estuary]

What is P?

[inner estuary]

So 1. and 2. here are showing that L is a vector space isomorphism?

[balmy linden]

Well P is the input

x l-> f(x) just refers to the map which associates f(x) to x

[inner estuary]

I'm also rlly lost where my orthonormal basis even comes into play, can't I prove those things without even knowing what the orthonormal basis for the inner product is?

[balmy linden]

Correct

[balmy linden]

For 1 and 2, you don't need orthogonality yet

for 3 you do

[inner estuary]

okay i will write out something for 1 and 2 and show u 😭

why this so hard lol

[balmy linden]

it's really not, I think you're just not absorbing information right for some reason

either I suck at explaining, either you're not paying attention

[inner estuary]

So this is our inner product isomorphism that we r trying to prove?

[balmy linden]

Yes

provided that you found (e1, e2) an orthonormal basis of P1(R)

[inner estuary]

okay ya so we have found (e1, e2) to be this:

[balmy linden]

Ok, that is fine

[inner estuary]

and we will make use of that for step 3. above?

[balmy linden]

Yes

You will need to know that e1 is orthogonal to e2 in P1(R)

that is, (e1, e2) is an orthonormal basis

[inner estuary]

okay so now my question is that, how did we know that this:

is the inner product isomorphism

like what definition told us it's this?

[balmy linden]

You can call it a very educated guess

it's just taking the coordinates of P in the basis (e1, e2)

[inner estuary]

So L is just mapping e1 and e2 to the standard basis vectors (1, 0) and (0,1) in R^2?

[balmy linden]

Yes

[inner estuary]

Ok I think I am slowly understanding , sorry for being so slow and difficult 😭

I'm writing out 1. and 2. above right now

[regal kiln]

it is not necessarily THE inner product isomorphism

it is an obvious candidate (because projections and so on..), but there can be others, so be careful with your wording

For context, P1 is a 2 dimensional vector space, so it is obvious that it is isomorphic to R2 as a vector space, the only remaining question is whether the inner product is preserved

you can think of it as a more general version of an isometry, where a map is an isometry if it preserves all inner squares, i.e, <f(x), f(x)> = <x,x>

when you map basis to basis, it follows immediately that the map is a vector space isomorphism

your only task now is to verify that inner product is preserved in the sense you linked before

[obtuse kayak]

Like i said before all stems from this

That is where L comes from

And (e_i) being orthonormal we can just say x_i=<x|e_i> for all i

So the map becomes x—>(<x|e_i>)_i

So there you have L, now like Rion said you need to prove that it satisfies the conditions (it’s an isomorphism and preserves inner product)

[inner estuary]

Ok gotcha, I will start with proving the first two conditions because I think I know how but the third condition and making using of the orthonormal basis I found is what i'm not sure about

[inner estuary]

Ok I have shown it's bijective and surjective aka one to one and onto

Not sure how to show L inner product preserving

I know we need to show (L(p)L(q)) = pq

But how to do so using my orthonornal basis I found?

[regal kiln]

any p is a linear combination of the basis you calculated, yes?

use that fact

[regal kiln]

you have verified the basis is orthonormal, yes?

[inner estuary]

I know my basis for the inner product is orthonornal ya

[inner estuary]

is this true?

[regal kiln]

what does orthonormality mean?

[inner estuary]

⟨v, w⟩ = 0 if v doesn't equal w

And ⟨v, v⟩ = 1 ?

[regal kiln]

you only have two elements in the basis

<e1,e2> = 0 and <e_i,e_i> = 1

[inner estuary]

Yeah

Okay lemme quickly write out how I have shown inner product is preserved

Not sure if this is right tho

[regal kiln]

sure, show us what you worked out

[inner estuary]

Ya almost done writing it, one sec

[inner estuary]

Is this good?

[regal kiln]

this is correct

the hardest part about this problem is to find an orthonormal basis

what your above argument shows is that any orthonormal basis will give you an isomorphism

[inner estuary]

gotcha

and since I also showed that this is linear, and bijective along with how I showed that L is inner product preserving, I have found AN inner product isomporphism to be L(P) and proved it fully now?

[inner estuary]

and that is what shows that L is inner product preserving right?

[regal kiln]

yes

L is an inner isomorphism, as required