Hello all!
So I am preparing for this maths exam and one of the topics is transformation. I have studied it, but I tend to get stuck at most of the questions. I need help with these:
1. How to identify what sort of transformation it is?
2. How to find the matrix of two combined transformations which which maps A onto B. ( what if ones reflection others enlargement or ones shearing and so on.)
I really need help. So if you think you can help me out, I'd be thankful.
I'm not exactly sure what you mean but I'll take a stab at it and maybe you can correct it if it is wrong:
It sounds like you are doing linear algebra (in which case you are posting in the wrong section) with something like
u = u0 + a (x - x0) + b (y - y0)
v = v0 + c (x - x0) + d (y - y0)
as a 'standard form' and writing this as
[(u-u0) (v-v0)] = [(x-x0) (y-y0)] A
which is another 'standard form' where A is the 2X2 matrix with row vectors (a b) and (c d). [Of course this could go on up to many dimensions rather than just the two given here where A is an nXn matrix.] If so, one way to describe this is that the point (x0, y0) in the Cartesian plane x-y is translated to the point (u0, v0) in the plane u-v [the translation part of the transformation], the vector [a b] determines the rotation and stretching/contraction for (x,y) to u. and [c d] determines the rotation and stretching/contraction for (x,y) to v. If
\(\displaystyle \frac{a}{\sqrt{a^2 + b^2}} = \frac{d}{\sqrt{c^2 + d^2}}\)
and
\(\displaystyle \frac{-b}{\sqrt{a^2 + b^2}} = \frac{c}{\sqrt{c^2 + d^2}}\)
then both u and v have been rotated through the same angle \(\displaystyle \theta\) where
\(\displaystyle cos(\theta) = \frac{a}{\sqrt{a^2 + b^2}} = \frac{d}{\sqrt{c^2 + d^2}}\)
and
\(\displaystyle sin(\theta) = \frac{-b}{\sqrt{a^2 + b^2}} = \frac{c}{\sqrt{c^2 + d^2}}\)
and the rotation is of the complete Cartesian x-y plane.
For question two, consider our transformation above and, for convenience, let u0=v0=x0=y0=0 so we can write
[u v] = [x y] A
and define two scalars \(\displaystyle \alpha\) and \(\displaystyle \beta \) and two angles \(\displaystyle \theta\) and \(\displaystyle \phi \) by
\(\displaystyle \alpha = \sqrt{a^2 + b^2}\),
\(\displaystyle \beta = \sqrt{c^2 + d^2}\),
\(\displaystyle cos(\theta) = \frac{a}{\alpha}\),
\(\displaystyle cos(\phi) = \frac{d}{\beta}\),
\(\displaystyle sin(\theta) = \frac{-b}{\alpha}\),
and
\(\displaystyle sin(\phi) = \frac{c}{\beta}\).
then our transformation can be written as
[u v] = [x y] C B
where the row vectors of B are [cos(\(\displaystyle \theta\)) -sin(\(\displaystyle \theta\))] and [sin(\(\displaystyle \phi\)) cos(\(\displaystyle \phi\))] and the row vectors of C are [\(\displaystyle \alpha\) 0] and [0 \(\displaystyle \beta\)]. If we let
[s t] = [x y] C
then
[u v] = [s t] B
Thus the original transformation can be thought of as, first, a contraction/stretching of x by \(\displaystyle \alpha\) to get s and a contraction/stretching of y by \(\displaystyle \beta\) to get t and then, if \(\displaystyle \phi = \theta\), a rotation of \(\displaystyle \theta\). Note that if \(\displaystyle \phi \ne \theta\), then u and v are not at right angles and the u-v plane is not a Cartesian plane.
'nuff for now
