Chapter 1
Some Linear Algebra
1.1 Introduction
In this chapter we recall elementary concepts of linear algebra necessary for the
development of the book. If the reader is interested in more details, we recommend
the textbooks [64, 137]; see also [13] for Theory of Modules which are, roughly
speaking, “vector spaces over rings”. The reader can consult the Appendix to recall
the concept of ring and field if necessary.
We begin by defining the concept of vector space, one of the main concepts of
linear algebra.
Definition 1.1.1 A vector space over a field K is a ordered triple (V, +, ·), where
V is a set,
+ : V × V −→ V
(u, v) −→ u + v
is an internal operation, called vector addition, and
· : K × V −→ V
(a, v) −→ av
is an external operation, called scalar multiplication, such that
(1) (V, +) is an abelian group:
(1.1) There exists 0 ∈ V such that v + 0 = 0 + v = v, ∀v ∈ V (zero or null vector);
(1.2) ∀ v ∈ V , there exists −v ∈ V such that v + (−v) = (−v) + v = 0 (symmetric);
(1.3) ∀ u, v, w ∈ V , u + (v + w) = (u + v) + w (associative);
(1.4) ∀ u, v ∈ V , u + v = v + u (commutative);
© Springer Nature Switzerland AG 2020
G. G. La Guardia, Quantum Error Correction, Quantum Science and Technology,
https://doi.org/10.1007/978-3-030-48551-1_1
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