The basis consisting of unit vectors is simple in that the coefficients in the representation are exactly the components of the vector With other types of bases it is not like that: the dependence of coefficients in
on for a general basis is not so simple.
Exercise 1. Put the basis vectors side by side, and write the vector of coefficients as a column vector. Then (1) becomes so that
The explicit formula from Exercise 1 shows, in particular, that the vector of coefficients is uniquely determined by and depends linearly on The coefficients of in another basis
may be different from those in (1). For future applications, we need to know how the coefficients in one basis are related to those in another. Put the basis vectors side by side, and write as a column vector.
Exercise 2. Let and be two bases in Then
Proof. With our notation (1) and (2) become and Thus, and (3) follows.
Definition 1. The matrix in (3) is called a transition matrix from to .
Changing bases to analyze matrices
Suppose we want to analyze Fix a basis and take any We can decompose as in (1). Then we have a vector of coefficients can be considered an original and - its reflection in a mirror or a clone in a parallel world. Instead of applying to we can apply it to its reflection to obtain To get back to the original world, we can use as a vector of coefficients of a new vector and call this vector an image of under a new mapping
The transition is unique and the transition is also unique, so definition (4) is correct.
Exercise 3. Show that
Proof. By Exercise 1, (4) can be written as Combining these two equations we get Since this is true for all the statement follows.
The point of the transformation in (5) is that may be in some ways simpler or better than Note that when we use the orthonormal basis of unit vectors, and
Definition 2. The matrix is called similar to