Part III
Linear Vector Spaces
In this part we treat vectors and their transformations in linear vector spaces so that
we can address various aspects of mathematical physics systematically but intuitively. We outline general principles of linear vector spaces mostly from an algebraic
point of view. Starting with abstract definition and description of vectors, we deal
with their transformation in a vector space using a matrix. An inner product is a
central concept in the theory of a linear vector space so that two vectors can be
associated with each other to yield a scalar. Unlike many of the books of linear
algebra and linear vector spaces, however, we describe canonical forms of matrices
before considering the inner product. This is because we can treat the topics in light
of the abstract theory of matrices and vector space without a concept of the inner
product. Of the canonical forms of matrices, Jordan canonical form is of paramount
importance. We study how it is constructed providing a tangible example.
In relation to the inner product space, normal operators such as Hermitian
operators and unitary operators frequently appear in quantum mechanics and electromagnetism. From a general aspect, we revisit the theory of Hermitian operators
that often appeared in both Parts I andII.
The last part deals with exponential functions of matrices. The relevant topics can
be counted as one of the important branches of applied mathematics. Also, the
exponential functions of matrices play an essential role in the theory of continuous
groups that will be dealt with in PartIV.