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
Proof. By the basis definition, runs over and therefore This implies The rest is obvious.
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 .
Matrix representation of a linear transformation
This topic in case of an orthonormal basis was considered earlier. Some people find the next general construction simpler.
Let be a basis and decompose as in (1). Let be a linear transformation. From
we see that the vectors uniquely determine . Decompose them further as Let us introduce a column-vector by Then the last equation takes the form With the matrix we can write one equation instead of
Combining (5), (6) and Exercise 1 we get Since is arbitrary, the linear transformation in the basis can be identified with the matrix
Definition 2. The matrix that is defined by (6) or (7) is called a representation of the linear transformation in the basis
Note two special cases: 1) if the basis is orthonormal, then is an orthogonal matrix and 2) when we use the orthonormal basis of unit vectors, and
Changing bases to analyze matrices
We want to see how the representations of a linear transformation in two bases are related to each other. For a basis summarized in the matrix we have (7). To reflect dependence of on the basis, let us denote it Then from (7) Similarly, for another basis we have The last two equations lead to Hence, Here is the transition matrix, so this can be written as
Definition 3. If there is a nonsingular matrix such that (8) is true, then the matrix is called similar to
Depending on the choice of the bases, one matrix may be simpler than the other.
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