How can mathematical induction be used to prove independence of linear vectors?

[transgalactic]

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 don't know how to find the number of times that
we switched each 2 lines

i don't 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 don't know how to use it on vectors??

 

[morphism]

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?

 

[transgalactic]

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 don't 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 don't understand how they got it??
the logic of this

 

[Defennder]

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.

 

[transgalactic]

for matrices 4X4 and 5X5 their is the same number of flips
(2)

so i am really puzzled about it