Simple example

Student request: change notations. Mine seem better than the book's, though. I think the books exposition (p207-210) is very confusing, partly by not using vector symbols to indicate vectors versus coordinates. I suggest you stick with my exposition.

To solve problems, it is often desirable or essential to change basis.

As an example, consider the vector of gravity . If I use a Cartesian coordinate system with the x-axis horizontal, the vector will be along the negative y-axis. I will call this coordinate system, (), the E-system.

Using the E-system, I can write the vector as:

In other words, the coordinates of vector in the E-coordinate system are and .

But if, say, the ground is under an angle with the horizontal, it might be much more convenient to use a coordinate system E^*, (), with the x-axis aligned with the ground:

In this new coordinate system, the coordinates of will be different. With a bit of trig, you see:

The coordinates of vector are now and

What if I need to change the coordinates of a lot of vectors from one coordinate system to the other? Is there a systematic way of doing this? The answer is yes; the following formula applies:

So the transformation of coordinates can be done by multiplying by a matrix P. This matrix consists of the basis vectors of the new coordinate system E^* expressed in terms of the old coordinate system E.

In particular,

and matrix P becomes:

Let's test it: P times the coordinates of vector in the E^*-system should give the coordinates in the E-system:

Multiplying out gives 0 and -g, which is exactly right.

Matrix P is called the transformation matrix from E to E^*. Note however that it really transforms coordinates in the E^*-system to coordinates in the E-system. You just have to get used to that language: a transformation matrix from A to B transforms B coordinates into A coordinates. No, I do not know who thought of that first.

What if you really want to transform E coordinates into E^* coordinates? No big deal: just multiply by the inverse matrix P^-1.