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February 23rd, 2009, 07:41 AM   #1
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Linear Alegebra of Equation Systems Minor Explanation Wanted

So, I'm going through the book "Linear Algebra Demystified" to start my re-entering in the world of mathematics. I have no previous experience in working with matrices but the first few pages seemed to explain the basics with pretty decent comprehension.

Thanks for reading this far, now comes the parts I don't yet comprehend fully.


Code:
M = [
2   -1  5
1  33  6
17  4  8
]
Now they tell me that I can practically do any of these three mutations to change the matrix rank.
Exchange two rows.
Replace a row with the scalar multiple of itself, as long as it's nonzero.
Replace one row by adding the scalar multipe of another row.

Now this last one I have quite a tough time figuring out. How can I do this when the matrix does not show results in the form of
Code:
2 -1 5 | 3*or something else
I figure something like this happens
Code:
2*M2 + M1 -> M1
M =
[
4  62  17
1  33  6
17  4  8
]
But of the three columns represents three unknown then doesn't that need to be in the calculation?

Code:
M =
[
4  62  17 | 2*(ans)
1  33  6
17  4  8
]
Also might someone want to explain how multiplications between an arbitrary and an elementary matrix really works. The book tells me it's simple. And I agree for 2x2 matrices but when it comes to 3x3 I'm quite lost.

Example
Code:
    
             7    ?2   3
A =          0    1    4   
            ?2   3    5

The elementary matrix that corresponds to 2R2 -> R2 is given by

            1    0    0
E1  =       0    2    0   
            0    0    1

E1A = 
         [ 1 0 0 ][ 7 -2 3]
         [ 0 2 0 ][ 0  1 4]
         [ 0 0 1 ][-2  3 5]
The calculations beyond that is what I do not understand. To me it seems like the second row of E1 is being multiplied with various versions of the first line in A. Why? The 7 is in every middle part.. like this

Code:
(0)(7) + (0)(0) + (0)(-2)  (0)(7)+(2)(1)+(0)(-2) etc
where does it get the first one from? it seems like E1,row 1 = 0 0 0 according to that.
and how does it actually work?

Some help would be greatly appreciated, and I'm sorry if I'm using the code tags in the wrong way.

vincenti
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February 23rd, 2009, 07:15 PM   #2
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What you've written has too many typos. If you quote a specific part of the book exactly, I can explain it.
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