Appendix C
Linear or Vector Spaces
This appendix presents properties of linear or vector spaces which are relevant to the
analysis of MDOF and continuous systems. It defines vector spaces, inner products
on these spaces, basis, and linear independence. The discussion is kept simple as it is
intended to develop intuition on these spaces and provide useful facts for dynamical
analysis. A rigorous treatment of these topics can be found in, e.g., [3].
Definition C.1 A linear or vector space V is a set that is closed to addition and
multiplication by scalar, i.e., if x and y are in V, then x + y and α x are in V, where
the scalar α is assumed to be a real number in our considerations. The operations
x + y and α x are performed component-by-component, i.e., the components of
x + y and α x are {x k + y k } and {α x k }, where {x k } and {y k }, k = 1, . . . , n, denote
the components of x and y.
Example C.1 The physical three-dimensional space, which is denoted by R 3 , is a
vector space. The elements x of this space can be represented by
x = x 1 i + x 2 j + x 3 k,
(C.1)
where the components x 1 , x 2 , and x 3 of x are the projections of this vector on the
unit vectors i, j, and k of the system of coordinates. If y is another three-dimensional
vector with components y 1 , y 2 , and y 3 , then x + y is also a three-dimensional vector
with components x 1 + y 1 , x 2 + y 2 , and x 3 + y 3 . Also, α x is a three-dimensional
vector with components α x 1 , α x 2 , and α x 3 .
The following two examples are extensions of the physical space, a threedimensional vector space, to vector spaces V with finite and infinite numbers of
components.
Example C.2 Suppose that the vectors x and y of the previous example belong to
an n-dimensional vector space R n , also referred to as the n-dimensional Euclidean
space. Denote, as previously, the unit vectors of this space by i k , k = 1, . . . , n. In
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
141
Linear or Vector Spaces
This appendix presents properties of linear or vector spaces which are relevant to the
analysis of MDOF and continuous systems. It defines vector spaces, inner products
on these spaces, basis, and linear independence. The discussion is kept simple as it is
intended to develop intuition on these spaces and provide useful facts for dynamical
analysis. A rigorous treatment of these topics can be found in, e.g., [3].
Definition C.1 A linear or vector space V is a set that is closed to addition and
multiplication by scalar, i.e., if x and y are in V, then x + y and α x are in V, where
the scalar α is assumed to be a real number in our considerations. The operations
x + y and α x are performed component-by-component, i.e., the components of
x + y and α x are {x k + y k } and {α x k }, where {x k } and {y k }, k = 1, . . . , n, denote
the components of x and y.
Example C.1 The physical three-dimensional space, which is denoted by R 3 , is a
vector space. The elements x of this space can be represented by
x = x 1 i + x 2 j + x 3 k,
(C.1)
where the components x 1 , x 2 , and x 3 of x are the projections of this vector on the
unit vectors i, j, and k of the system of coordinates. If y is another three-dimensional
vector with components y 1 , y 2 , and y 3 , then x + y is also a three-dimensional vector
with components x 1 + y 1 , x 2 + y 2 , and x 3 + y 3 . Also, α x is a three-dimensional
vector with components α x 1 , α x 2 , and α x 3 .
The following two examples are extensions of the physical space, a threedimensional vector space, to vector spaces V with finite and infinite numbers of
components.
Example C.2 Suppose that the vectors x and y of the previous example belong to
an n-dimensional vector space R n , also referred to as the n-dimensional Euclidean
space. Denote, as previously, the unit vectors of this space by i k , k = 1, . . . , n. In
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Switzerland AG 2021
M. D. Grigoriu, Linear Dynamical Systems,
https://doi.org/10.1007/978-3-030-64552-6
141