1. Algebraic Structures

Definition 1.1: Group

A group is a set $G$ together with a binary operation $*: G \times G \to G$ satisfying:

(G1) Associativity: $(a * b) * c = a * (b * c)$ for all $a, b, c \in G$
(G2) Identity: There exists $e \in G$ with $e * a = a * e = a$ for all $a \in G$
(G3) Inverses: For each $a \in G$ , there exists $a^{-1} \in G$ with $a * a^{-1} = a^{-1} * a = e$

Example 1.1: Standard Examples of Groups

$(\mathbb{Z}, +)$ — integers under addition (abelian, infinite)
$(\mathbb{Q}^*, \cdot)$ — nonzero rationals under multiplication (abelian)
$(\mathbb{Z}_n, +)$ — integers mod $n$ under addition (abelian, finite)

Definition 1.2: Ring

A ring is a set $R$ with two binary operations $+$ (addition) and $\cdot$ (multiplication) such that:

(R1) $(R, +)$ is an abelian group
(R2) Multiplication is associative: $(ab)c = a(bc)$
(R3) Distributive laws: $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$

Definition 1.3: Field

A field is a set $F$ with operations $+$ and $\cdot$ such that:

(F1) $(F, +)$ is an abelian group with identity $0$
(F2) $(F \setminus \{0\}, \cdot)$ is an abelian group with identity $1$
(F3) Distributivity: $a(b + c) = ab + ac$
(F4) $0 \neq 1$

Example 1.2: Standard Fields

$\mathbb{Q}$ — rational numbers
$\mathbb{R}$ — real numbers
$\mathbb{C}$ — complex numbers
$\mathbb{Z}_p$ for prime $p$ — finite field with $p$ elements

Theorem 1.1: ℤₚ is a Field

$\mathbb{Z}_p$ is a field if and only if $p$ is prime.

2. Complex Numbers

Definition 2.1: Complex Numbers

The complex numbers $\mathbb{C}$ is the set $\mathbb{R} \times \mathbb{R}$ with operations:

Addition: $(a, b) + (c, d) = (a + c, b + d)$
Multiplication: $(a, b) \cdot (c, d) = (ac - bd, ad + bc)$

We write $i = (0, 1)$ and identify $a \in \mathbb{R}$ with $(a, 0) \in \mathbb{C}$ . Then $(a, b) = a + bi$ .

Definition 2.2: Complex Conjugate and Modulus

For $z = a + bi \in \mathbb{C}$ :

The complex conjugate is $\bar{z} = a - bi$
The modulus is $|z| = \sqrt{a^2 + b^2}$

Theorem 2.1: Euler's Formula

For any $\theta \in \mathbb{R}$ :

e^{i\theta} = \cos\theta + i\sin\theta

Every non-zero complex number can be written as $z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$ .

3. Equivalence Relations

Definition 3.1: Equivalence Relation

An equivalence relation on a set $S$ is a relation $\sim$ that is:

Reflexive: $a \sim a$ for all $a \in S$
Symmetric: If $a \sim b$ , then $b \sim a$
Transitive: If $a \sim b$ and $b \sim c$ , then $a \sim c$

Definition 3.2: Equivalence Class

The equivalence class of $a \in S$ is:

[a] = \{b \in S : a \sim b\}

Theorem 3.1: Equivalence Classes Partition the Set

The equivalence classes of an equivalence relation partition $S$ : they are disjoint and their union is $S$ .

4. Gaussian Elimination

Definition 4.1: Elementary Row Operations

The three elementary row operations (EROs) on a matrix are:

Row swap (Rᵢ ↔ Rⱼ): Interchange rows $i$ and $j$
Row scaling (Rᵢ ← cRᵢ): Multiply row $i$ by a non-zero scalar $c$
Row addition (Rᵢ ← Rᵢ + cRⱼ): Add $c$ times row $j$ to row $i$

Definition 4.2: Row Echelon Form

A matrix is in row echelon form (REF) if:

All zero rows are at the bottom
The leading entry (pivot) of each non-zero row is to the right of the pivot in the row above
All entries below a pivot are zero

Definition 4.3: Reduced Row Echelon Form

A matrix is in reduced row echelon form (RREF) if:

It is in row echelon form
Every pivot is equal to 1
Each pivot is the only non-zero entry in its column

Theorem 4.1: Uniqueness of RREF

Every matrix has a unique reduced row echelon form.

5. Vector Spaces

Definition 5.1: Vector Space

A vector space over a field $F$ is a set $V$ together with two operations:

Vector addition: $+: V \times V \to V$
Scalar multiplication: $\cdot: F \times V \to V$

satisfying the following eight axioms for all $u, v, w \in V$ and $\alpha, \beta \in F$ :

Axiom V1: Additive Commutativity

u + v = v + u

Axiom V2: Additive Associativity

(u + v) + w = u + (v + w)

Axiom V3: Additive Identity

There exists $0 \in V$ such that $v + 0 = v$ for all $v \in V$ .

Axiom V4: Additive Inverse

For each $v \in V$ , there exists $-v \in V$ such that $v + (-v) = 0$ .

Axiom V5: Multiplicative Identity

1 \cdot v = v

Axiom V6: Scalar Associativity

\alpha(\beta v) = (\alpha\beta)v

Axiom V7: Distributivity over Vector Addition

\alpha(u + v) = \alpha u + \alpha v

Axiom V8: Distributivity over Scalar Addition

(\alpha + \beta)v = \alpha v + \beta v

Example 5.1: The Coordinate Space Fⁿ

For any field $F$ and positive integer $n$ , the set $F^n = \{(a_1, \ldots, a_n) : a_i \in F\}$ is a vector space with:

(a_1, \ldots, a_n) + (b_1, \ldots, b_n) = (a_1 + b_1, \ldots, a_n + b_n)

\alpha(a_1, \ldots, a_n) = (\alpha a_1, \ldots, \alpha a_n)

Example 5.2: Polynomial Space F[x]

The set $F[x]$ of all polynomials with coefficients in $F$ is a vector space with standard addition and scalar multiplication.

Example 5.3: Matrix Space M_{m×n}(F)

The set of all $m \times n$ matrices with entries in $F$ is a vector space with entry-wise operations.

Theorem 5.1: Basic Properties of Vector Spaces

In any vector space $V$ over a field $F$ :

The zero vector is unique
Additive inverses are unique
$\alpha \cdot 0 = 0$ for all $\alpha \in F$
$0 \cdot v = 0$ for all $v \in V$
$(-1) \cdot v = -v$ for all $v \in V$
$\alpha v = 0 \implies \alpha = 0 \text{ or } v = 0$

Proof of Theorem 5.1 (Property 3):

Let $\alpha \in F$ .

\alpha \cdot 0 = \alpha \cdot (0 + 0) = \alpha \cdot 0 + \alpha \cdot 0

Adding $-(\alpha \cdot 0)$ to both sides gives $0 = \alpha \cdot 0$ .

∎

Frequently Asked Questions

Why do we need all eight vector space axioms?

Each axiom captures an essential property. Remove any one, and the theory breaks: without additive inverses, we can't subtract; without the distributive law, scalar multiplication and addition are unrelated. The axioms are the minimal set ensuring a coherent algebraic structure.

What's the difference between a group and a field?

A group has one operation (like addition) with identity and inverses. A field has two operations (addition and multiplication) where both operations form abelian groups (with the exception that 0 has no multiplicative inverse). Fields are the scalars of linear algebra.

Why is the field important for vector spaces?

The same set can be a vector space over different fields with different properties. ℂ over ℂ is 1-dimensional; ℂ over ℝ is 2-dimensional. The field determines what 'scalar' means and affects dimension and structure fundamentally.

Can a vector space have just one element?

Yes! The trivial vector space {0} contains only the zero vector. It's a valid vector space (all axioms are satisfied vacuously or trivially). Its dimension is 0.

What's the connection between Gaussian elimination and vector spaces?

Gaussian elimination is the computational tool for working with vector spaces. It finds bases, computes dimensions, solves systems, and determines linear independence. The row space, column space, and null space are all computed using Gaussian elimination.

Foundations & Vector Spaces Practice

10

Questions

0

Correct

0%

Accuracy

1

Which of the following is NOT a group under addition?

Easy

Not attempted

2

What is the identity element in the group

(\mathbb{Z}, +)

?

Easy

Not attempted

3

Which of the following is a field?

Medium

Not attempted

4

What is

i^{2023}

?

Easy

Not attempted

5

Which axiom fails for

\mathbb{N}

under standard addition and scalar multiplication by reals?

Easy

Not attempted

6

What is the zero vector in the vector space of

2 \times 2

real matrices?

Easy

Not attempted

7

In the vector space

\mathbb{R}[x]

of polynomials, what is

-p(x)

for

p(x) = x^2 + 3x - 1

?

Easy

Not attempted

8

Which is NOT an elementary row operation?

Easy

Not attempted

9

What is the modulus of

z = 3 + 4i

?

Easy

Not attempted

10

In a vector space, can the zero vector have a non-zero scalar multiple?

Medium

Not attempted