linear algebra Gram-Schmidt in Function Spaces problem

**Ackbeet** · August 18th 2010, 10:49 AM

Whoops. Not sure integration by parts will get you the $H_{1}$ integral. Let me think on that one a bit more.

**SpiffyEh** · August 18th 2010, 10:51 AM

so the normalized H_0 would be sqrt(sqrt(pi))?
Oh ok, this problem is a lot more work than I was expecting.

**Ackbeet** · August 18th 2010, 11:07 AM

Correct on the $H_{0}$ normalization!

For the $H_{1}$ normalization, I would do the same dirty trick as for $H_{0}$ : square the integral. You're gonna get the following:

Let
$\displaystyle{I=\int_{-\infty}^{\infty}(2x)^{2}e^{-x^{2}}\,dx}.$

Then

$\displaystyle{I^{2}=16\int_{-\infty}^{\infty}x^{2}e^{-x^{2}}\,dx\int_{-\infty}^{\infty}y^{2}e^{-y^{2}}\,dy}$

$\displaystyle{=16\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}x^{2}y^{2}e^{-(x^{2}+y^{2})}\,dx\,dy.}$

Switching to polar, we get

$\displaystyle{I^{2}=16\int_{0}^{2\pi}\int_{0}^{\in fty}r^{4}\cos^{2}(\theta)\sin^{2}(\theta)e^{-r^{2}}\,r\,dr\,d\theta}$

$\displaystyle{=16\int_{0}^{2\pi}\cos^{2}(\theta)\s in^{2}(\theta)\,d\theta\int_{0}^{\infty}r^{4}e^{-r^{2}}\,r\,dr.}$

The angle integral succumbs to standard Calc II techniques, perhaps using double-angle formulas. For the integral on the right, substitute $u=r^{2}$ and THEN integrate by parts twice.

Yeah, this is a lot of work! Welcome to special functions.

**SpiffyEh** · August 18th 2010, 11:19 AM

Wow, I really hope nothing like this is on the exam. I tried it with wolframapha just to see what would happen and got 2sqrt(pi) from that integral. so sqrt(2sqrt(pi)) would be the normalized H_1? Sorry for not going through the steps, i'm trying to understand as much as I can before the exam in a couple hours

**Ackbeet** · August 18th 2010, 11:47 AM

WolframAlpha certainly does seem to compute things correctly. That is the correct norm, I believe. So that does it for 6.3, right?

I understand not going through the steps, if you're going for the big picture. You know better than me whether you need the big picture or the details.

So how about 6.4?

**SpiffyEh** · August 18th 2010, 11:50 AM

Hopefully the big picture is all I need. We only had 3 1 hour long class periods to go over 2 chapters so he really can't make things way too difficult on the exam. Well, at least I hope he thinks that way.

6.4 - I'm a little lost about doing this method with functions rather than vectors. I understand it with vectors.

**Ackbeet** · August 18th 2010, 12:06 PM

The procedure is precisely the same as for vectors, except that instead of a dot product, you do integrals, and instead of plugging vectors into the dot product, you're plugging functions into the integrals. It's a fair amount of integration, as you can see. Does that help?

**SpiffyEh** · August 18th 2010, 12:18 PM

Yes, i'll go through and try to do it and show you what I get if you don't mind checking it.

**Ackbeet** · August 18th 2010, 12:22 PM

I don't mind checking in the least. Fire away!

**Ackbeet** · August 19th 2010, 04:31 PM

How did your test go?

**SpiffyEh** · August 19th 2010, 09:25 PM

Sorry for not getting back sooner, I tried to get on but it would never load, maybe it was my connection. The test was pretty bad, I messed up somewhere with the arithmetic somewhere but it was too late to turn back which made it even harder. But besides that I got the actual concepts. As long as I got above a 78% I'll keep my A

**Ackbeet** · August 20th 2010, 02:26 AM

Yeah, the MHF was down because of problems with the LaTeX Preview feature. It was taking forever to preview LaTeX code output. Now that's fixed (yay!).

Well, here's hoping you did well on your test. If you got the concepts, and it wasn't a multiple choice (which, in my very strong opinion, are an abomination), then surely partial credit would help you out, assuming your teacher does that.

Was this thread helpful for your understanding?

Have a good one!

**SpiffyEh** · August 20th 2010, 10:17 AM

Yeah, it should be ok, luckly it wasn't multiple choice. The thread was very helpful. Thank you

**Ackbeet** · August 20th 2010, 10:18 AM

You're welcome. Have a good one!

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