How Do Matrices Influence Forms and Dimensions in Geometry?

[drag]

Greetings !

Can anyone please explain to me, preferably with as
little math and as much theory as possible , how
matrices work in terms of forms and dimensions and what's
the relevant mechanism ?

I mean, I used to program Star Trek spaceships for a computer
game I once made using matrices. But, nobody ever explained
to me how and why it works, those were just meaningless tables
with sprecific relevant operators to me. I also know the
whole addition/multiplication/other operations stuff(though
I can hardly remember it now), but not how and why it works for geometrical forms in various dimensions and stuff.

Thanks !

Live long and prosper.

 

[HallsofIvy]

You are asking a big question!

You might want to look at:

http://www.sosmath.com/matrix/matrix.html

 

[Guybrush Threepwood]

OK so maybe this is a response for a restricted area of matices use but here goes.
I program in OpenGL and I think the other 3D programming stuff has the same principles. To make a 3D scene you have to use coordinate transformation. These transformations (translation, rotation, scaling) can be written as a linear sistem between the coordinates in the old reference system, and the coordinates in the new reference system. Any linear equation system can be expressed in a matriceal form. So this is one of the reasons you use matrices. Also basic matrix operations are easy to program.

 

[STAii]

Or, in other words ...
You can do all the world transformation without matrices, it is only that matrices is a convinient way to save points, data, and transformations.
(i learned that while trying to learn Direct3D)

 

[plus]

One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.

 

[Sting]

One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.

Yep, Matrix multiplication is noncommutative and I know some people who failed exams because of it (even though the teacher had mentioned many times).

 

[Guybrush Threepwood]

Originally posted by plus
One of the most interesting things about matrices are than, due to the method of multiplication A * B is not necessarily the same as B * A.

true, true...
that's why in OpenGL a rotation, followed by a translation doesn't have the same result as a translation followed by a rotation. To do that you have to make some matrix multiplication and since it's not commutative the results are dependent of the order in which you multiply.