5.1 Algebraic Structures

abstract algebra We’ll get to linear algebra in a minute. First, it’s worthwhile to emphasize what algebra actually is. Here, we’re not talking about the elementary algebra you learned in school, which is about manipulating variables (read: arithmetic with letters). We’re talking about the abstract algebra of modern mathematics, which is about studying algebraic structures.

algebraic structure

An algebraic structure consists of three things: sets of elements, operations over those sets, and the axioms those operations satisfy. Though mathematicians have developed whole zoos of exotic algebraic structures, we’re interested in simpler structures that often involve only a single set of elements and only binary operations.

group

For example, a group, is a set, $G$ , equipped with a binary operation, $(\cdot):~G \times G \to G$ , that satisfies the following four properties:

Closure: $\id{closure}{\forall a,\ b \in G: a \cdot b \in G}$ .
Identity: $\id{identity}{\exists e \in G:\forall a \in G,\ a \cdot e = e \cdot a = a}$ .
Associativity: $\id{associativity}{\forall a,\ b,\ c \in G: a \cdot (b \cdot c) = (a \cdot b) \cdot c}$ .
Invertibility: $\id{invertibililty}{\forall a\in G,\ \exists a^{-1} \in G:\ a\cdot a^{-1} = a^{-1} \cdot a = e}$ .

abelian group An abelian group is a group whose elements are also commutative:

\t{5.1}{\forall a,\ b \in A: a \cdot b = b \cdot a.}

Example 5.1 (Symmetries of the Square)

Consider the group of rotations of a square, $C_4$ . We have four elements: the identity, a rotation by $\pi/2$ , a rotation by $\pi$ , and a rotation by $3\pi/2$ . The identity is the identity element, and the rotations are inverses of each other. The group is closed under composition, and the composition is associative. The group is Abelian, because the rotations commute with each other.

If we conside all symmetries of a square (rotations and reflections), $D_4$ ,¹ we introduce four new elements: two reflections across the vertical/horizontal axes and two reflections across the diagonal axes. The group is still closed under composition, and the composition is still associative. However, the group is no longer abelian because of the reflections.

	$e$	$r$	$r^2$	$r^3$	$t_x$	$t_y$	$t_{AC}$	$t_{BD}$
$e$	$e$	$r$	$r^2$	$r^3$	$t_x$	$t_y$	$t_{AC}$	$t_{BD}$
$r$	$r$	$r^2$	$r^3$	$e$	$t_{AC}$	$t_{BD}$	$t_y$	$t_x$
$r^2$	$r^2$	$r^3$	$e$	$r$	$t_y$	$t_x$	$t_{BD}$	$t_{AC}$
$r^3$	$r^3$	$e$	$r$	$r^2$	$t_{BD}$	$t_{AC}$	$t_x$	$t_y$
$t_x$	$t_x$	$t_{BD}$	$t_y$	$t_{AC}$	$e$	$r^2$	$r^3$	$r$
$t_y$	$t_y$	$t_{AC}$	$t_x$	$t_{BD}$	$r^2$	$e$	$r$	$r^3$
$t_{AC}$	$t_{AC}$	$t_x$	$t_{BD}$	$t_y$	$r$	$r^3$	$e$	$r^2$
$t_{BD}$	$t_{BD}$	$t_y$	$t_{AC}$	$t_x$	$r^3$	$r$	$r^2$	$e$

Cayley table This is called a Cayley table, and it’s a useful way to visualize the structure of a finite group. Entry $(r,c)$ is what you get after applying the $c$ th operation to the $r$ th element. We can see this isn’t an abelian group because the table is not symmetric. subgroup Additionally, we can see that $C_4$ is a subgroup of $D_4$ (it’s self-contained in the top-left quadrant of the table).

Exercise 5.1 Prove that

C_4

is an Abelian group.

Exercise 5.2 Prove that

D_4

is a non-Abelian group.

Exercise 5.3 Which of the following are/are not groups? Which are Abelian?

ring, lattice

Moving up, rings and lattices are sets equipped with two binary operations that are commutative and associative. The difference between these two is how the two operations distribute.

For rings, the operations (“addition” and “multiplication”) are distributive:

\t{5.2}{\forall a,\ b,\ c \in R: a \cdot (b + c) = a \cdot b + a \cdot b.}

For lattices, the operations (“join” and “meet”) are absorptive:

\t{5.3}{\forall a,\ b \in L: a \wedge (a \vee b) = a \vee (a \wedge b) = a.}

Exercise 5.4 Prove that

(\{\text{True}, \text{False}\}, \vee, \wedge)

is a lattice, where

\vee

and

\wedge

have the standard Boolean interpretation (“and,” “xor”)

field

Finally, fields are rings with an additional property: every element has an inverse. That is, $\forall a \in F,\ \exists a^{-1} \in F:\ a \cdot a^{-1} = a^{-1} \cdot a = 1$ . They’re the structure you’re most familiar with, as they’re the basis of arithmetic. The rational numbers, $\QQ$ , are a field, as are the real numbers, $\RR$ , and the complex numbers, $\CC$ .

5.2 Vector Spaces

vector space For this chapter, we’re mainly interested in a particular kind of algebraic structure called a vector space. Vector spaces are defined over two existing structures:

An Abelian group over a set, $V$ , of “vectors”, whose operation is called vector addition ( $+_V: V \times V \rightarrow V$ ) and whose identity element is the zero vector, usually denoted $\b{0}$ .
A field over a set, $F$ , of “scalars” with all the usual arithmetic operations ( $\cdot_F$ , $+_F$ , $-_F$ , $1$ , $0$ ). Usually, this field is the real numbers, $\mathbb{R}$ , but it could be any field.

A vector space combines these structures through an operation called scalar multiplication ( $\cdot_S: V \times F \rightarrow V$ ) that distributes over vector and field addition. That is, for all $\b v,\ \b w \in V$ and for all $a,\ b \in F:$

\t{5.4}{a \cdot_S (\b v +_V \b w) = a \cdot_S \b v +_V a \cdot_S \b w,}

and

\t{5.5}{(a +_F b) \cdot_S \b v = a \cdot_S \b v +_V b \cdot_S \b v.}

Moreover, scalar multiplication is “compatible” with field multiplication,

\t{5.6}{\forall \b v \in V,\ \forall a,\ b \in F: (a \cdot_F b) \cdot_S \b v = a \cdot_S (b \cdot_S \b v),}

and there identity of the field addition operation is the identity of the scalar multiplication operation,

\t{5.7}{\forall \b v \in V: 1 \cdot_S \b v = \b v.}

To simplify the notation (at the expense of more ambiguity), we almost always drop the subscripts, and we omit the scalar multiplication operation when it’s clear from context. So, we can write:

\t{5.8}{(a + b) (\b v + \b w) = a \b v + a \b w + b \b v + b \b w,}

Be careful to not confuse vector addition, $+_V$ , with field addition, $+_F$ or scalar multiplication, $\cdot_S$ , with field multiplication, $\cdot_F$ .

Example 5.2 (Types of vectors)

Geometric vectors. Often, vectors are introduced as arrows. Two arrows can be added together by adding their endpoints. Multiplying an arrow by a scalar $\lambda$ is equivalent to stretching (or squeezing) the arrow by a factor of $\lambda$ .
Polynomial vectors. Polynomials are also vectors. Adding two polynomials together is equivalent to adding their coefficients. Multiplying a polynomial by a scalar $\lambda$ is equivalent to multiplying each coefficient by $\lambda$ .
Elements of $\RR^n$ . On a more abstract level, tuples of $n$ numbers are vectors. For example, the vector $\vec{v} = (1, 2, 3)$ is an element of $\mathbb{R}^3$ . If we add a second vector (element-wise) $\vec{w} = (4, 5, 6)$ , we obtain a third vector $\vec{v} + \vec{w} = (5, 7, 9)$ which is also in $\mathbb{R}^3$ . If we multiply $\vec{v}$ by a scalar $a = 2$ , we get another vector $2 \cdot \vec{v} = (2, 4, 6)$ .

In fact, both geometric vectors and polynomial vectors can be seen as special cases of elements of $\RR^n$ . Given a basis of coordinates, we can represent a geometric vector as the tuple of numbers representing its endpoint. Similarly, given a set of polynomials, we can represent a polynomial vector as the tuple of numbers representing its coefficients.

Insert picture here

Exercise 5.5: Prove that

\mathbb{R}^3

is a vector space (with vector addition & scalar multiplication defined element-wise.)

To prove that $\mathbb{R}^3$ is a vector space, we need to show that $\mathbb{R}^3$ is an Abelian group under vector addition, and that element-wise multiplication by a scalar satisfiess the properties of scalar multiplication.

Altogether, this involves ten axioms:

Axiom	Operation	Meaning
Closure 1	Vector addition	$\b v + \b w \in \mathbb{R}^3$
Associativity	Vector addition	$(\b v + \b w) + \b x = \b v + (\b w + \b x)$
Commutativity	Vector addition	$\b v + \b w = \b w + \b v$
Identity 1	Vector addition	$\b v + \b 0 = \b v$
Inverse	Vector addition	$\b v + \b v^{-1} = \b 0$
Closure 2	Scalar multiplication	$a\b v \in \mathbb{R}^3$
Distributivity 1	Scalar multiplication	$a (\b v + \b w) = a \b v + a \b w$
Distributivity 2	Scalar multiplication	$(a + b) \b v = a \b v + b \b v$
Compatibility	Scalar multiplication	$(a \cdot b) \b v = a (b \b v)$
Identity 2	Scalar multiplication	$1 \b v = \b v$

5.3 Matrices

5.4 Systems of Linear Equations

5.5 Linear Independence, Basis, and Rank

5.6 Linear mappings

5.7 Further Reading

For more on algebraic structures, check out Julie Morunuki’s cheatsheet.

For more on linear algebra, check out MML, Strang (2003), [Golan (2007)], Hogben (2013), Axler (2015), Liesen and Mehrmann (2015), Pavel Grinfeld’s online series, Gilbert Strang’s notes, and 3Blue1Brown’s videos.

This is called the dihedral group of degree four. ↩

6 Linear Algebra

Authors

Last Updated

5.1 Algebraic Structures

Example 5.1 (Symmetries of the Square)

5.2 Vector Spaces

Example 5.2 (Types of vectors)

5.3 Matrices

5.4 Systems of Linear Equations

5.5 Linear Independence, Basis, and Rank

5.6 Linear mappings

5.7 Further Reading

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6 Linear Algebra

Authors

Last Updated

5.1 Algebraic Structures

Example 5.1 (Symmetries of the Square)

5.2 Vector Spaces

Example 5.2 (Types of vectors)

5.3 Matrices

5.4 Systems of Linear Equations

5.5 Linear Independence, Basis, and Rank

5.6 Linear mappings

5.7 Further Reading

Footnotes

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