Matrix Transformation

Davio

Hiya, just a quick question regarding matrices:

The following 2 column vectors are a particular form of transformation when applied to 2x2 matrices:

(1) (0)
(0) (1)

Am I right in saying the first, takes just the x component of the matrice, and the second the y component? Thing is, To me that sounds odd because the matrice's given are just matrices, with nothing to indicate they may be vectors or whatever, they seem to be just arrays of numbers.

jhicks

You're thinking too hard. [1 0; 0 1] x [a b; c d] = ? (I've delimited rows by ; )

Davio

If I do what you wrote, you just get the same matrix again, but if you do it by my column vectors you get only half the matrix..

jimvoit

The matrix M={{1,0},{0,1}} is an identity matrix. When it multiplies a matrix
N={{a,b},{c,d}} the result is M N=N. In this case you can interpret the rows of the
identity matrix as unit vectors i, j in the x and y directions respectively. Then M N takes the dot product of i and j with the columns of N, thus resulting in the original N.

Davio

Ah i see, what I meant is, (1, 0 ) being one vector, which is then multiplied by M. ie. M.I, which gives just a rotation through i? I can see when my 2 column vectors are combined they equal an identity matrix, however I only want, one of the columns at any time, to see what they do to my matrix.
M={1,0}, N={{a,b},{c,d}} mn= a,b ... what actually is this called? ie. its only part of the original matrix, I think the answer given is, its a rotation of some sort, along the x axis.

jimvoit

If I understand the question as first stated, you are asking for an
interpretation of the results of the matrix multiplication I N, where
I=
1 0
0 1

and N=

a b
c,d

The result of applying the rules of matrix multiplication is I N=N, that is why I is called
an identity matrix.

As for an interpretation of the multiplication, the dot product of the first row of I
with the first column of N gives a, and so on.

Incendentially, there is no rotation here.

slider142

MN is not a valid matrix product, unless you meant M^TN. NM is also a valid matrix product. The map f(N) = NM just selects the first column of the matrix N, the image of the first basis vector. The map f(N) = M^TN maps N to the image of the first basis vector of the dual space in the dual of the image of N. If N is a rotation matrix, you will note that the result is a rotation of the basis vector.

Davio

AH, nope sorry, I wasn't being clear, I mean, I have 2 different column vectors, (which happen to look just liike an identity matrix when put together)
Slider142 's explanation is kinda of what i'm looking for, except I don't quite understand it.
I did mean:(1 ). M. By mapping image, do you mean, the vector is placed along 1 ?
(0 ) 0