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#1 2007-08-09 16:27:55

mikau
Member
Registered: 2005-08-22
Posts: 1,504

linear algebra is kicking my face in

teaching myself a crash course in linear algebra over the summer. It was going fine until i got to the chapter on vector spaces. I'm able to understand the theorems , examples and solutions concerning them (with some difficulty in some cases) but I'm very rarely able to solve a problem without looking at the solution. I understand the solution, but when doing the problem it just doesn't come to me. I'm getting frusterated because no two problems seem at all similar and i can't seem to devise any approach to tackling them. sad

I used to think I was good at math but this is starting to make me feel like i'm terrible at it!

I think, if anything, i really need more practice problems, and perhaps some pointers for what to look for when I get stuck. If any member here is willing to offer me a few excersizes, i'd be most greatful.

What I'm working on now is vector spaces, subspaces, spanning sets, and basis sets.  Particularly, finding a basis of a vector space.

And if anyone can give some general advice i'd appreciate that as well.


A logarithm is just a misspelled algorithm.

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#2 2007-08-10 16:11:37

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: linear algebra is kicking my face in

You mean you always make mistakes during the calculation?

Then you have to watch all the entries (usually 3) in one row and imagine doing the scaler product simultaneously.

Last edited by George,Y (2007-08-10 16:17:10)


X'(y-Xβ)=0

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#3 2007-08-10 16:21:13

George,Y
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Registered: 2006-03-12
Posts: 1,379

Re: linear algebra is kicking my face in


X'(y-Xβ)=0

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#4 2007-08-11 06:26:33

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: linear algebra is kicking my face in

no i don't have trouble with the formulas and computations.

The problems that get me are, say, find a basis of this matrix, find a basis that contains these entries, etc. Find  a basis for the intersecttion of these two sets, etc.

I don't know if there is some formula developed that allows you to find a basis directly but at this point the problem sets are asking me to find them so i think i should be able.

Let me try to be more specific. What is the best first approach to finding a basis, should you just guess at a linearly independant set? What  if you need to find a basis that contains another set, or a basis of a union, or intersection set?


A logarithm is just a misspelled algorithm.

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#5 2007-08-12 05:57:58

HallsofIvy
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Re: linear algebra is kicking my face in

mikau wrote:

no i don't have trouble with the formulas and computations.

The problems that get me are, say, find a basis of this matrix, find a basis that contains these entries, etc. Find  a basis for the intersecttion of these two sets, etc.

I don't know if there is some formula developed that allows you to find a basis directly but at this point the problem sets are asking me to find them so i think i should be able.

Let me try to be more specific. What is the best first approach to finding a basis, should you just guess at a linearly independant set? What  if you need to find a basis that contains another set, or a basis of a union, or intersection set?

Then you seem to be having more problems than you recognize: for one thing a MATRIX does not have BASIS.  Only vector spaces have bases.

   You are correct that to find a basis for a set spanned by given vectors, you want to find a linearly independent set.  But it should not be necessary to "guess" that set.  One method is to use the vectors as rows of a matrix (which may be where you got that first reference to a matrix), then do "row reduction".  Any rows that turn into all zeros were originally dependent on the others and the non-zero rows that remain are the basis vectors.

#6 2007-08-12 14:55:04

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: linear algebra is kicking my face in

Then you seem to be having more problems than you recognize: for one thing a MATRIX does not have BASIS.  Only vector spaces have bases.

A matrix produces a vector space by what you mentioned below.  Is it that much of a leap in logic to assume that this is exactly what mikau meant?

mikau, your book should give you steps to produce such results, even if they are vauge.  You are going to have to be more specific on what you are having trouble on.  However, the one thing to do in linear algebra is to write everything out based on the definitions, even if that means things will become very ugly.  Linear Algebra isn't meant to be pretty.


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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#7 2007-08-12 18:37:05

George,Y
Member
Registered: 2006-03-12
Posts: 1,379

Re: linear algebra is kicking my face in

"mikau, your book should give you steps to produce such results"

I want to say the same words. The procedure you want is only routine, and your book should just contain it.


X'(y-Xβ)=0

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#8 2007-08-12 21:21:37

mikau
Member
Registered: 2005-08-22
Posts: 1,504

Re: linear algebra is kicking my face in

how bout that, they've given me no procedure. For simpler vector spaces, its usually east to guess, (like a basis of  R^3 is (1,0,0), (0,1,0) and (0,0,1) but for more complicated sets, or when they get creative, it can be hard.

whats really getting me is not the computational problems but the problems where they ask you to prove some random fact and you have no idea where to begin.

here was one that got me good.

In the vector space R^4 let
A = Span { (1, 2, 0, 1), (-1, 1, 1, 1)}
B = Span{(0,0,1,1), (2,2,2,2)}

determine the intersection of A and B and compute its dimension.

They never once showed me how to find an intersection of any type of set before.

and Ricky, what exactly do you mean write everything out based on defintions?

Last edited by mikau (2007-08-12 21:23:46)


A logarithm is just a misspelled algorithm.

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#9 2007-08-14 05:31:42

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: linear algebra is kicking my face in

and Ricky, what exactly do you mean write everything out based on defintions?

The first thing to do with that problem is write out in vector notation every single element of both A and B:

For A:

a - b
2a + b
b
a + b

For B:

2d
2d
c + 2d
c + 2d

Now what does it mean for an a vector to be in both spaces?


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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