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  #11  
Old 11-18-2005, 03:03 AM
jason_t jason_t is offline
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Default Re: linear algebra

[ QUOTE ]


Homework assignment: what is a commutative division ring usually called? [img]/images/graemlins/wink.gif[/img]

[/ QUOTE ]

Answer in white: <font color="white">A field</font>.

Homework assignment: prove that every finite integral domain is a field.
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  #12  
Old 11-18-2005, 07:20 AM
pzhon pzhon is offline
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Default Re: linear algebra

[ QUOTE ]
I'm going to be taking linear algebra over the winter intersession at my college this winter. Basically it is a semester divided by 3, so the class is accelerated. What can I do to prepare for the class itself?

-2nd year mathematics major

[/ QUOTE ]
There are several types of introductory linear algebra classes. Most of the advice people have given seems to assume you are getting one particular type which emphasizes simple but tedious calculations. Other types include introductions to rigorous proofs and introductions to numerical methods. That you are taking the class in the intersession increases the chance you are getting one of the latter two types. Hybrids are possible.

At some point in your studies as a mathematics major, the emphasis changes from being able to compute things and knowing what is true to handling abstract systems and being able to prove things. Some colleges make this change in a class on number theory, some in real analysis, some in abstract algebra, and some in linear algebra. If this class is taught as an introduction to proofs, it would be a big mistake to focus on the calculations during the class, even if it is easiest to test your ability to calculate things. Before the class, review determinants, matrix multiplication, and Cramer's Rule from high school algebra so these don't get in the way. During the class, spend extra time making sure you know exactly what the definitions are of the objects studied, what properties have been established, and how these were proved. In fact, I recommend keeping a separate notebook to record these (and no calculations).

If the class is taught as an introduction to applied mathematics, I recommend familiarizing yourself with the syntax of the computational system used in the class (perhaps Matlab, Maple, or Mathematica) or the programming language suggested.

If this is an ordinary class on linear algebra, buy lots of paper and some quality pens.
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  #13  
Old 11-18-2005, 08:39 AM
chezlaw chezlaw is offline
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Default Re: linear algebra

[ QUOTE ]
[ QUOTE ]
Homework assignment: what is a commutative division ring usually called? [img]/images/graemlins/wink.gif[/img]

[/ QUOTE ]

Answer in white: <font color="white"> A field.</font>

[/ QUOTE ]

Another homework assignment:

What's purple and commutes?

Answer in white: <font color="white"> An abelian grape.</font>

chez
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  #14  
Old 11-18-2005, 10:53 AM
Indiana Indiana is offline
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Default Re: linear algebra

Only thing you need to know about Algebra is that the subgroup of a cyclic group is cyclic. Think about it.

Indy
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  #15  
Old 11-18-2005, 11:42 AM
Isura Isura is offline
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Default Re: linear algebra

[ QUOTE ]
Homework assignment: prove that every finite integral domain is a field.

[/ QUOTE ]

Harder: Prove that every finite division ring is a field. [img]/images/graemlins/grin.gif[/img]
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  #16  
Old 11-18-2005, 12:03 PM
Isura Isura is offline
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Default Re: linear algebra

[ QUOTE ]
I'm going to be taking linear algebra over the winter intersession at my college this winter. Basically it is a semester divided by 3, so the class is accelerated. What can I do to prepare for the class itself? I believe it has an emphasis on matrices, but I'm not sure. My math grades are decent mostly A's and B's and that is why I'm attempting the class, what advice can anyone give.

-2nd year mathematics major

[/ QUOTE ]

Does your school offer a theoretical linear algebra course? This is what you would want to take as a 2nd year math major. Linear algebra comes up a lot in higher studies of mathematics, and it is essential that you have a sound foundation in the subject. For an outline of what a theoretical linear algebra course would cover, search for the book "Linear algebra done right" by Axler on amazon.com. IMO it's the best linear algebra book out there. An old but classical book on the subject is "Finite-dimensional vector spaces" by Paul Halmos.

Even if your course doesn't go into that kind of depth, it is worthwhile to read (and do the problems) in either of these books. Linear algebra is really important not only in mathematics, but in numerical/applied math, computer science (eg. graphics) and various other scientific areas. So even if you don't want to go to grad school, it's worthwhile to learn the subject properly now.
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  #17  
Old 11-18-2005, 01:55 PM
jason_t jason_t is offline
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Default Re: linear algebra

[ QUOTE ]
[ QUOTE ]
Homework assignment: prove that every finite integral domain is a field.

[/ QUOTE ]

Harder: Prove that every finite division ring is a field. [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

I was building up to that one! The one I stated actually isn't that hard. An interesting thing is that last night I couldn't remember the result I stated above but I could remember the idea behind its proof and was able to reconstruct the result.
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  #18  
Old 11-18-2005, 03:50 PM
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Default Re: linear algebra

[ QUOTE ]
What's purple and commutes?

Answer in white: <font color="white"> An abelian grape.</font>

[/ QUOTE ]

What's purple, commutes, and is worshipped by a limited number of people?

Answer in white: <font color="white"> a finitely venerated abelian grape.</font>
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  #19  
Old 11-18-2005, 03:53 PM
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Default Re: linear algebra

[ QUOTE ]
[ QUOTE ]
Homework assignment: prove that every finite integral domain is a field.

[/ QUOTE ]

Harder: Prove that every finite division ring is a field. [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

Proof by citation: Wedderburn, 1905. [img]/images/graemlins/wink.gif[/img]
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  #20  
Old 11-18-2005, 05:10 PM
jason_t jason_t is offline
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Location: Another downswing?
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Default Re: linear algebra

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Homework assignment: prove that every finite integral domain is a field.

[/ QUOTE ]

Harder: Prove that every finite division ring is a field. [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

Proof by citation: Wedderburn, 1905. [img]/images/graemlins/wink.gif[/img]

[/ QUOTE ]

I'm still mystified every time I think about this result. Finiteness =&gt; commutativity. It's absurd when you think about it. There is a truly beautiful proof of this result in Proofs from the Book.
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