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#1 2005-09-22 16:02:26

kybasche
Member
Registered: 2005-09-22
Posts: 8

Matrix Algebra - Reflection of Vector and Leading one's of rref matrix

Hey all, lovely forum!  I've got 2 problems, both seem to be pretty straight forward, but I'm getting stuck.  I've used the notation for matrices that my calculator accepts... a comma seperates values in the same column, and a colon denotes the start of a new row, I also seperated rows with spaces to make it a bit easier to read

1)  (this is paraphrased from Linear Algebra w/ Applications by Otto Bretscher)

two nxm matrices in reduced row-echelon form are the the same type if they have the same number of leading 1's in the same locations... i.e. [*1,2,0 : 0,0,*1] and [*1,3,0 : 0,0,*1] --- stars indicate corresponding leading 1's

How many types of 3x2 matrices in rref are there?

I get 3... [1,a : 0,0 : 0,0]    [1,0 : 0,1 : 0,0]  and the [0,0 : 0,0 : 0,0]
However, the book's answer is that there are 4 types... am I missing something??

2)   (again paraphrased from the same book)

consider matrix A = [a,b : b,-a]  where a^2+b^2=1

Find two nonzero perpendicular vectors 'v' and 'w' such that A*v=v and A*w=-w
Solve for the vectors in terms of 'a' and 'b'

For this one, the matrix 'A' is a reflection transformation, and in order for the reflection of 'v' to be equal to 'v,' I would imagine that 'v' has to be parallel to the line about which the reflection is taking place.  I can't seem to find a way to get to the answer though.

The answer is given in the book as

v = [b : (1-a)]
w = [-b : (1+a)]



Sorry that this was so long smile  Any help on either of these would be very much appreciated.  Thanks a bunch!

~Derek

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#2 2005-10-05 09:56:12

kybasche
Member
Registered: 2005-09-22
Posts: 8

Re: Matrix Algebra - Reflection of Vector and Leading one's of rref matrix

For the first problem, I think the one I was missing was

[0,1:0,0:0,0]

Anyone have thoughts on the second problem?

~Derek

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#3 2005-10-20 06:30:13

priya
Guest

Re: Matrix Algebra - Reflection of Vector and Leading one's of rref matrix

i have got modueration exam coming up and i can't find the work that i need. i'm doing foundation work at the moment i really need your help.

Thanks,
bye

#4 2005-10-20 09:18:59

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,529

Re: Matrix Algebra - Reflection of Vector and Leading one's of rref matrix

"modueration exam" ? smile

Anyway, tell us what you need, we may be able to point you in the right direction.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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