RyozKidz said:
a given matrix of 2X2 is \stackrel{A}{C} \stackrel{B}{D}
can anyone pls tell me wat can be told by this matrix?
or is this a coordinates of a vector like a row vector or column vector??
And how does this matrix of 2X2 come from without any linear equation??
Matrices really come from the idea of vector spaces. A matrix is a "representation of a linear transformation on a vector space, in a given system of coordinates". That's a bit of a mouthful, so in simplified terms:
Imagine your ordinary 2D plane with a cartesian system of co-ordinates. Any point (x,y) can be represented by a position vector from the origin to that point. So you can think of a plane as a set of points or a set of position vectors - let's do the latter.
A transformation of a vector space is just a rule that maps any vector to another vector in the space. So you could imagine a transformation that rotates all vectors clockwise by 37 degrees. Or one that adds (3,5) to each vector. Or one that doubles each vector. And so on.
Of all possible transformations we could consider, a
linear transformation is one with a couple of special properties. Let's call the transformation A(), the vectors a, b, c and so on, and use m,n,... for real numbers (I'm too lazy to use LateX). Then a linear transformation is one for which A(a + b) = A(a) + A(b) for all vectors, and A(ma) = mA(a) for all vectors and scalars. In words, 'the transform of a sum of vectors is the sum of the transforms of each vector' and 'the transform of a scalar multiple of a vector is the scalar multiple of the transform of the vector'.
Now here's what a matrix is: it's a way of representing a particular linear transform in a particular set of coordinates. Suppose we know that a linear transform takes i to mi + nj, and takes j to pi + qj (where i and j are the standard unit vectors along the x and y axes). Then we can write that transform as a matrix:
(m n)
(p q)
And multiplying any vector by that matrix (on the left, of course) now gives the transform of that vector. You can see this by doing the matrix multiplication explicitly, writing a general vector as, say, ri + sj, and using the linearity properties mentioned above.
Often matrices are first introduced to students as just 'tables of numbers'. This is all fine if you just want to crunch through numbers, and the definitions of matrix addition and multiplication by a scalar are 'intuitively obvious'. But then they teach you about matrix multiplication, and the obvious response is 'why on Earth is it such a complicated, apparently made-up rule'? And the answer is to do with what matrices really are: the definition of matrix multiplication comes out of considering them as transformations of vectors. It also helps when they introduce you to determinants, identity and inverse matrices.
