2.1 Change of Basis

Let us reconsider the vector x=(2, 1, 3). Fully written out, it is

in a given Cartesian ^[1] frame e _i, i=1, 2, 3. Suppose we appoint a new frame , i=1, 2, 3, such that

(From these expansions we could calculate the and verify that they are non-coplanar.) Recalling that x is an objective, frame-independent entity, we can write

In these calculations it is unimportant whether the frames are Cartesian; it is important only that we have the table of transformation

It is clear that we can repeat the same operation in general form. Let x be of the form

(2.1)

with the table of transformation of the frame given as

Then

Here have introduced a new notation, placing some indices as subscripts and some as superscripts. Although this practice may seem artificial, there are some fairly deep reasons for following it.

2.2 Dual Bases

To perform operations with a vector x we must have a straightforward method of calculating its components-ultimately, no matter how advanced we are, we must be able to obtain the x ⁱ using simple arithmetic. We prefer formulas that permit us to find the components of vectors using dot multiplication only; we shall need these when doing frame transformations, etc. In a Cartesian frame the necessary...