7.6   COORDINATE TRANSFORMATIONS

It has been mentioned that vector geometry is usually introduced with the aid
of three orthogonal unit vectors: i, j, and k.

For now, let us keep to two dimensions and consider just (x,y)′, meaning
xi + yj.

Now suppose that there are two sets of axes in action. With respect to our
first set the point is (x,y)′ but with respect to a second set it is (u,v)′. Just how
can these two vectors be related?

What we have in effect is one pair of unit vectors i, j, and another pair,
l, m, say. Since both sets of coordinates represent the same vector, we
have

Now each of the vectors l and m must be expressible in terms of i and j.
Suppose that

or in matrix form

We want the relationship in this slightly twisted form, because we want to
substitute into

to eliminate vectors l and m to get

Now the ingredients must match:

Although this exercise is now graced with the name “vector geometry,” we
are merely adding up mixtures in just the same form as the antics in the candy
store.

To convert our (u,v)′ coordinates into the (x,y)′ frame, we simply multiply
the coordinates by an appropriate matrix that defines the mixture.

Suppose, however, that we are presented with the values of x and y and are
asked to find (u,v)′. We are left trying to solve two simultaneous equations:

In traditional style, we multiply the top equation by d and subtract c times
the second equation to obtain

and in a similar way, we find

which we can rearrange as

where the constant 1/(ad − bc) multiplies each of the coefficients inside the
matrix.

If the original relationship between (x,y)′ and (u,v)′ was

then we have found an “inverse matrix” such that

The value of (ad − bc) obviously has special importance—we will have great
trouble in finding an inverse if (ad − bc) = 0. Its value is the “determinant”
of the matrix T.