Re: While I'm calculating Determinant; What I'm doing exactly.

On Mar 9, 12:03 pm, Nimo <azeez...@xxxxxxxxx> wrote:
On Mar 9, 9:54 pm, José Carlos Santos <jcsan...@xxxxxxxx> wrote:



Nimo wrote:
' What exactly is Determinant ? '

I know it is related to the concepts of Linear Algebra.
I'm trying to understand the concepts of
' Linear Algebra ' myself from this beautiful book.

" Elements of Abstract and Linear Algebra "

by

E.H. Connell
Department of Mathematics
University of Miami
P.O. Box 249085
Coral Gables, Florida 33124 USA
e...@xxxxxxxxxxxxxxx
___________

lets take an example

2 apples + 3apples = 5 apples
here at the end I got 5 apples
as a result.

like wise;

if I perform the below operation for these

2, 1, 4 & 5 apples.

What I did exactly. what is the end result.

A = 2 1
4 5

=> |A| = 6; what is 6 here ?

Consider the linear map f:R^2 ---> R^2 defined by

f(x,y) = (2x + y,4x + 5y).

and consider the any bounded region A of the plane. Then the absolute
value of the determinant (6, in your case) means that the area of f(A)
will be six times the area of A.

The sign of the determinant tells you whether _f_ preserves orientation
(when the sign is +) or not (when the sign is -).

Best regards,

Jose Carlos Santos

Fantastic..!
that's really helpful.

If you have time; Can you please give me the
official definition with an example
for the determinant; I'll take note's of it.

but, a small doubt
(may be weird to you)

R => line
R^2 => Area
R^3 => Volume
R^4 => ?
.
.
.
R^n => ?? { what should I call; when n > 3 }

so long
nimo

Why should I refuse a good dinner
simply because I don't understand
the digestive processes involved?

by Oliver Heaviside.

There are many ways to define and calculate determinants, probably you
would be best to read http://en.wikipedia.org/wiki/Determinant as it
has a summary of most methods including worked examples. When proving
theorems involving determinants, I prefer to use the Leibniz formula,
as it is one of the most compact, elegant descriptions. In
applications, if you just need to compute the value of some particular
determinant, the Laplace (or cofactor) expansion is often used, or
Gaussian elimination. There is also a perfectly good definition
hinted to above that is great from the perspective of having an
intuition of what we are talking about: Namely, given any multilinear
(linear in all variables) map f:R^N --> R^N, the determinant of f is
(+/-) the measure of f(A) where A is any set of measure 1. The
positive sign is selected if the map f is orientation preserving, and
the negative selected if the map is orientation reversing. To answer
your other question then about how to proceed in the sequence: length,
area, volume, ..., some people would say in the 4th dimension that it
would be hypervolume, but it is quite customary to continue calling it
volume, or hypervolume well after the 3rd dimension if no confusion is
possible. Alternately, all of these things are examples of 'measures'
on spaces, so you could call them all measures, or to be perfectly
precise "the (lebesgue) measure on R^N".
.

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