Defining an Inner Product on the Vector Space of Polynomials in x

[Zurtex]

My question is:

Let [itex]V = \mathbb{R}_1 [x][/itex] be the vector space of polynomials in x of degree at most 1. For [itex]f(x) \, , \, g(x) \in \mathbb{R}_1 [x][/itex], define:

[tex]<f(x) \, , \, g(x)> \, = \int_0^1 x^2 f(x) g(x) dx[/tex]

Show that this defines an inner product on [itex]\mathbb{R}_1[x][/itex]. (You may assume the result which says that for function [itex]h(x) \, , \text{if} \, h(x) \geq 0 \, \, \forall x[/itex], then [itex]\int_0^1 h(x) dx \geq 0[/itex] with equality iff [itex]h \equiv 0[/itex])


I have no idea how to even approach this, can someone point me in the right direction please.

 

[matt grime]

What are the requirements for < , > to be an inner product?
There are 4 rules

<u,v>=<v,u>
<u+v,w>=<u,w>+<v,w>
<ku,v>=k<u,v> for k a scalar
<u,u> is strictly positive unless u=0 when <0,0>=0

and they are all easily seen to be satisfied

 

[Zurtex]

Oh right, cool thanks. Sorry the course in Linear Algebra I took the lecturer made it sound far more complex than it was. I've had to go and work it out all myself and then teach it to the rest of the class.

 

[mathwonk]

every proof begins the same way: what is the definition, i.e. the meaning of the property you are supposed to verify? here it is "inner product". all matt has done is give you the definition.

If you know even this much about proofs you could have begun this problem. If you do not know this, it is hard to believe it the fault of your linear algebra lecturer, as this is far more basic than that.

It is more likely the fault of your high school geometry course, or whatever course in proving things was omitted by your high school so they could cram in AP calculus I imagine.

Or, heavens, could it be your own fault? nahhh.

 

[quetzalcoatl9]

<u,u> is strictly positive unless u=0 when <0,0>=0

it may be useful for the OP to know that this last rule, of nondegeneracy, is not always defined for an inner product. an example would be the metric of special relativity.

also, it is sometimes phrased as:

if for some fixed vector [itex]\vec{v}[/itex], [tex] <\vec{v}, \vec{w}> = 0[/tex] for all [tex]\vec{w}[/tex] then [tex] \implies \vec{v} = 0[/tex]

 

[matt grime]

In a linear algebra course inner products are non-degenerate, otherwise it is just a bilinear map.

 

[Zurtex]

every proof begins the same way: what is the definition, i.e. the meaning of the property you are supposed to verify? here it is "inner product". all matt has done is give you the definition.

If you know even this much about proofs you could have begun this problem. If you do not know this, it is hard to believe it the fault of your linear algebra lecturer, as this is far more basic than that.

It is more likely the fault of your high school geometry course, or whatever course in proving things was omitted by your high school so they could cram in AP calculus I imagine.

Or, heavens, could it be your own fault? nahhh.

I wasn't taught the a particularly good definition of an inner product in my Linear Algebra course, the definition I was given by Matt was enough for me to start. So please don't assume.

I've yet to come across a mathematics which if I have time for I can't prove all the theorems myself, however being in the middle of 6 exams and it only being a small part of a question that may or may not come up and having to teach various people the course all over again it's not something I'm going to spend more than 3 minutes looking at.

 

[Don Aman]

it may be useful for the OP to know that this last rule, of nondegeneracy, is not always defined for an inner product. an example would be the metric of special relativity.

All inner products are nondegenerate, even the inner product of special relativity. You're getting nondegeneracy confused with positive-definiteness.

 

[mathwonk]

next time you need something like the definition of inner product, you might try the index of your book, or searching the internet. it came up immediately on mathworld, exactly what matt grime gave you, in about 1 second.

 

[matt grime]

Damn, my secret is out. That's exactly what I did. Not because I do not know the definition, but because i want to check that it is easily available. I find it quite depressing that the commonest phrase I type is "google for 'subject' and include the word wolfram". 9 times out of ten it gives you the answer (for the kinds of questions asked here).

 

[mathwonk]

It also saves typing if you copy it over. I doubt if exposing the secret will diminish the number of questions here whose answers are readily available.

Its like basketball, everyone knows playing defense wins games, but few want to do it.

by the way i notice a legal disclaimer on your answer. my employers have recently also begun to request them at least for our official websites. i guess the intent is to inform the gullible or litigious that opinions expressed on the interenet are just what they would obviously appear to be, opinions.

Also to distinguish official sources of information from unofficial ones. I think this is doomed by the size of the job though, since already there are official websites of the employers themselves carying wholly outdated information.

by the way matt, you once expressed a motivation to provide a ready reference for answers to FAQ. perhaps a ready reference for resources would be helpful and more feasible. It might include such gems as looking in ones textbook, say at the index or table of contents, as well as more appealing internet based ones.

Or basic suggestions such as: when trying to prove X is a Y, read the definitions of X and Y, and compare them.

 

[Zurtex]

next time you need something like the definition of inner product, you might try the index of your book, or searching the internet. it came up immediately on mathworld, exactly what matt grime gave you, in about 1 second.

I normally do look things up on MathWorld first, I can't explain why I didn't this time I just don't remember it coming to mind. Also I have no book on Linear Algebra or any other maths topic for that matter.

 

[mathwonk]

free books:

http://joshua.smcvt.edu/linearalgebra/

http://www.math.miami.edu/~ec/book/

http://mathforum.org/library/topics/linear/

http://dmoz.org/Science/Math/Publications/Online_Texts/

http://www.math.miami.edu/~ec/book/

http://www.math.uga.edu/~roy/

 

[matt grime]

I chose to put in a disclaimer since I was now using my official work web address in my .sig. As it has the word "maths" in the link I don't want people thinking that I am doing anything in any official capacity. I agreee that shouldn't be necessary but it's reasonable advice these days when common sense has ceased to be something we can rely on. On the odd occasions I post on Usenet to say sci.math(.research) I never use my work email (and not just because I don't want spam). There have been cases of some usenet posters having their employers contacted because of the posts they made to usenet. I even had the wonderful Doron Shadmi threaten to contact my department since he didn't like the comments I made about his work, despite the fact he emailed me privately to ask for said comments.


My web thing where I explain a little about "how i prove things" has, as its first piece of advice, "make sure you understand the definitions of all the terms involved", followed by (paraphrasing from memory) "look at your notes and see if it is similar to something stated there; perhaps you can modify the proof".

 

[HallsofIvy]

Many years ago, a student in a Linear Algebra Class (although this might happen in any course) asked me, at the beginning of class, how to do a particular homework problem. I glanced at the problem, pointed to one word in the problem, and asked "what's the definition of this word?". Since he couldn't give me an answer, I looked around at the other students and asked "Does anyone here know the definition?". Again getting no answer, I said "I think we've found the difficulty" and sat down! The students stared at me while I sat there not saying or doing anything. Eventually, more and more student's started leafing through the textbook for the definition and finally, one looked the word up in the index! After he had written the definition of that particular word (and I no longer remember what it was), it was easy for the class to do the homework problem.

It always amazes me that student would think they could do a problem when they don't understand the very words in the problem. Of course, often the student has some vague idea of what the word means and doesn't (yet!) realize that the exact words of a definition are crucial. In mathematics definitions are "working definitions"- you use the precise words of the definitions in proofs and doing problems.

 

[mathwonk]

Halls, you sound like a very good teacher. I admire your courage.

 

[HallsofIvy]

I once told a class "the most important thing I can teach you is to learn without a teacher" and they replied "you certainly are good at that"!

I'm sure they meant it as a compliment!

 

[mathwonk]

your classes are also courageous! I think they learned it from you.