Wikipedia:Reference desk/Archives/Mathematics/2015 May 29

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May 29

Math puzzle.

I'm stumped! Who can help. This is from a past test paper.

Let V and W be finite dimensional real vector spaces. Define the rank rT and nullity nT of a linear transformation T : V → W. State and prove the rank nullity theorem.

Let V1, V2, V3, V4 be finite dimensional spaces and Ti

Vi → Vi+1, i = 1, 2, 3, linear

maps.

Suppose that T1 is injective (one-to-one), T3 is surjective (onto), the image of T1 equals the kernel of T2 and the image of T2 equals the kernel of T3.

Show that (i) dim V2 = dim V1 + rT2 , (ii) dim V1 + dim V3 = dim V2 + dim V — Preceding unsigned comment added by 82.28.140.226 (talk) 17:05, 29 May 2015 (UTC)

I don't know this stuff, but I've taken the liberty of adding wikilinks to your post as a first step for both of us. (Usually we don't edit each others' comments but I hope this is a tolerable deviation) Wnt (talk) 17:18, 29 May 2015 (UTC)

This question is #2 from Oxford's 2010 First P.E. for Pure Mathematics. -- ToE 00:34, 30 May 2015 (UTC)

Gambler problem...

A gambler has an initial fortune of size k, where k is a non-negative integer. He plays a sequence of independent games such that on each play he wins 1 with probability p, or loses 1 with probability p, or his fortune remains unchanged with probability q, where 2p + q = 1. He decides to stop playing when either his fortune reduces to zero or his fortune reaches m, where m > k. Let N(k) be the number of games played until he stops.

How can I find EN(k)?

Now suppose, that his initial fortune is a random variable X, where X is binomially distributed with parameters m and α. What now is the expected number of games played? — Preceding unsigned comment added by 82.28.140.226 (talk) 17:07, 29 May 2015 (UTC)

You may wish to start by reading and understanding the mathematics used in Random walk#One-dimensional random walk, as that is in many ways similar to this problem. After you have digested that (as well as the articles which Wnt has linked in your question above), get back to us with the progress you have made and let us know if there is anywhere you are stuck. -- ToE 19:55, 29 May 2015 (UTC)

There was a similar question at Wikipedia:Reference desk/Archives/Mathematics/2014 July 19. The upshot is that you can solve this type of problem by setting up a system of linear equations. Not sure if this is homework, but the question seems to be copied from [1] which makes it dubious from a COPYVIO standpoint. --RDBury (talk) 23:09, 29 May 2015 (UTC)

There can't be copyright violation in the quotation of a short excerpt like the OP did. See Fair Use (Wikipedia goes by U.S. laws); I assume de minimis also allows some sort of scholarly quotation. I have quoted larger bits of text on occasion to answer a question, and will continue to do so. Wnt (talk) 00:23, 30 May 2015 (UTC)

WP:Non-free content requires that brief verbatim textual excerpts from copyrighted media be properly attributed, which the OP did not do. Perhaps our subsequent links to the sources are sufficient. This question was #3 from Oxford's 2006 First P.E. for Pure Mathematics. -- ToE 00:48, 30 May 2015 (UTC)