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2 question in lenear algebra

  1. Jan 4, 2008 #1
    1.
    i posted the first question on

    http://img151.imageshack.us/my.php?image=98702238hn9.jpg

    the problem is to find the deteminant for this big matrices

    i got the solution and i was told that in order to solve it ,we
    need to switch lines
    and for every switch of line we multiply the matrices by (-1)
    the problem is that i dont know how to find the number of times that
    we switched each 2 lines

    i dont know how they get the expresion bellow

    2.
    http://img242.imageshack.us/my.php?image=87397722lz2.jpg
    my second proble is to make a proof in lenear algebra

    the question is:

    if V1,V2.....Vn are independent vectors ,not equaled to zero
    of a lenear operator of T
    that belong to personal values lamda1,lamda2.....lamda n
    (lamda numbers are the rootes of the polinomial that we get from the matrices ).

    then we need to proove that the vector of
    V1,V2.....Vn

    are independant lenearly vectors

    i tried to understand the proove there
    its not finished
    it is a proove using mathematical induction

    i solved using mathematical induction many mathematical equations
    i dont know how to use it on vectors??
     
  2. jcsd
  3. Jan 4, 2008 #2

    morphism

    User Avatar
    Science Advisor
    Homework Helper

    For the determinant one I'm assuming you're trying to get it into upper triangular form? Well, there's a very primitive way you can flip the matrix upside down. First lift the bottom row by switching it with the row above it -- so far we have 1 move. Next lift it once again, and lift the (n-1) row so that it stays beneath it -- so far we have 1+2 moves. And so on. If you can't follow what I'm saying, try it out with a small matrix. Hopefully this will make it clearer.

    For the second question, why don't you show us what you have so far?
     
  4. Jan 5, 2008 #3
    regarging the first
    question:
    i know that in a 3X3 matrices we need 1 flip
    in a 4X4 matrices we need 2 flips
    but i dont know what is the link between the nuber
    of flips and the size of the matrices.

    regarging the second question:
    i showed the partial solution to the problem
    i dont understand how they got it??
    the logic of this
     
  5. Jan 5, 2008 #4

    Defennder

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    Homework Helper

    For the first question, try it out for larger matrices like 5x5 and 6x6, 7x7 matrices. You'll notice 2 patterns for nxn matrices when n is odd and n is even. The number of flips needed for an nxn matrix depends on whether it is odd or even.
     
  6. Jan 5, 2008 #5
    for matrices 4X4 and 5X5 their is the same number of flips
    (2)

    so i am realy puzzled about it
     
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