Page 396 - 35Linear Algebra

396 Movie Scripts

Then we could write this as

1 2 3 4 1 2 3 4
=
σ(1) σ(2) σ(3) σ(4) 3 2 4 1
We could write this permutation in two steps by saying that first we swap 3
and 4, and then we swap 1 and 3. The order here is important.

This is an even permutation, since the number of swaps we used is two (an even
number).

Elementary Matrices

This video will explain some of the ideas behind elementary matrices. First
think back to linear systems, for example n equations in n unknowns:
 1 1 1 2 1 n 1
 a x + a x + · · · + a x = v
1
2
n

 2 1 2 2 2 n 2
 a x + a x + · · · + a x = v

1
n

2
.
.
 .




a x + a x + · · · + a x = v .
 n 1 n 2 n n n
n
1
2
We know it is helpful to store the above information with matrices and vectors
 1 1 1   1   1 
a a · · · a x v
1 2 n
2
2
 a 2 a 2 · · · 2  x   v 
 1 2 a 

n 



M :=  . . . , X :=  . , V :=  . .
 . . .  .   . 
.
.
.
.
. 
a n a n · · · a n x n v n
2
n
1
Here we will focus on the case the M is square because we are interested in
its inverse M −1 (if it exists) and its determinant (whose job it will be to
determine the existence of M −1 ).
We know at least three ways of handling this linear system problem:
1. As an augmented matrix

M V .
Here our plan would be to perform row operations until the system looks
like
I M −1 V ,
(assuming that M −1 exists).
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