Chapter 11
Vectors and Their Transformation
In this chapter, we deal with the theory of finite-dimensional linear vector spaces.
Such vector spaces are spanned by a finite number of linearly independent vectors,
namely, basis vectors. In conjunction with developing an abstract concept and
theory, we mention a notion of mapping among mathematical elements. A linear
transformation of a vector is a special kind of mapping. In particular, we focus on
endomorphism within a n-dimensional vector space V
n . Here, the endomorphism is
defined as a linear transformation: V
n
! V
n . The endomorphism is represented by a
(n, n) square matrix. This is most often the case with physical and chemical
applications, when we deal with matrix algebra. In this book we focus on this type
of transformation.
A non-singular matrix plays an important role in the endomorphism. In this
connection, we consider its inverse matrix and determinant. All these fundamental
concepts supply us with a sufficient basis for better understanding of the theory of
the linear vector spaces. Through these processes, we should be able to get
acquainted with connection between algebraic and analytical approaches and gain
a broad perspective on various aspects of mathematical physics and related fields.
11.1 Vectors
From both fundamental and practical points of view, it is desirable to define linear
vector spaces in an abstract way. Suppose V is a set of elements denoted by a, b, c,
etc. called vectors. The set V is a linear vector space (or simply a vector space), if a
sum a + b 2 V is defined for any pair of vectors a and b and the elements of V satisfy
the following mathematical relations:
a þ b
ð
Þþc ¼ a þ b þ c
ð
Þ,
ð11:1Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_11
433
Vectors and Their Transformation
In this chapter, we deal with the theory of finite-dimensional linear vector spaces.
Such vector spaces are spanned by a finite number of linearly independent vectors,
namely, basis vectors. In conjunction with developing an abstract concept and
theory, we mention a notion of mapping among mathematical elements. A linear
transformation of a vector is a special kind of mapping. In particular, we focus on
endomorphism within a n-dimensional vector space V
n . Here, the endomorphism is
defined as a linear transformation: V
n
! V
n . The endomorphism is represented by a
(n, n) square matrix. This is most often the case with physical and chemical
applications, when we deal with matrix algebra. In this book we focus on this type
of transformation.
A non-singular matrix plays an important role in the endomorphism. In this
connection, we consider its inverse matrix and determinant. All these fundamental
concepts supply us with a sufficient basis for better understanding of the theory of
the linear vector spaces. Through these processes, we should be able to get
acquainted with connection between algebraic and analytical approaches and gain
a broad perspective on various aspects of mathematical physics and related fields.
11.1 Vectors
From both fundamental and practical points of view, it is desirable to define linear
vector spaces in an abstract way. Suppose V is a set of elements denoted by a, b, c,
etc. called vectors. The set V is a linear vector space (or simply a vector space), if a
sum a + b 2 V is defined for any pair of vectors a and b and the elements of V satisfy
the following mathematical relations:
a þ b
ð
Þþc ¼ a þ b þ c
ð
Þ,
ð11:1Þ
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_11
433