Unit 4: Fundamentals of Abstract Algebra and Linear Algebra

1. Abstract Algebra: Groups, Rings, and Fields

1.1 Groups

A group is a fundamental algebraic structure consisting of a set equipped with a binary operation that combines any two of its elements to form a third element. To be a group, the set and its operation must satisfy four axioms:

• Closure: For all elements a, b in G, the result of the operation a * b is also in G.
• Associativity: For all elements a, b, c in G, (a * b) * c = a * (b * c).
• Identity Element: There exists an element e in G such that for every element a in G, e * a = a * e = a.
• Inverse Element: For every element a in G, there exists an inverse element a^-1 in G such that a * a^-1 = a^-1 * a = e.

Definition of a Group: A non-empty set G with a binary operation * is called a group (G, *) if it satisfies closure, associativity, existence of an identity element, and existence of inverse elements. If the operation is also commutative (a * b = b * a), it is called an Abelian group.

Example: The set of integers ℤ under addition (+) is an Abelian group.

• Closure: a + b ∈ ℤ for all a, b ∈ ℤ.
• Associativity: (a + b) + c = a + (b + c) for all a, b, c ∈ ℤ.
• Identity Element: 0 ∈ ℤ, and 0 + a = a + 0 = a for all a ∈ ℤ.
• Inverse Element: For every a ∈ ℤ, -a ∈ ℤ such that a + (-a) = (-a) + a = 0.

1.2 Rings

A ring is an algebraic structure that generalizes integers, equipped with two binary operations, typically called addition and multiplication. It's an Abelian group under addition, and multiplication is associative and distributive over addition.

Definition of a Ring: A set R with two binary operations, addition (+) and multiplication (×), is called a ring (R, +, ×) if it satisfies the following axioms:

• (R, +) is an Abelian group.
• Multiplication is associative: (a × b) × c = a × (b × c) for all a, b, c ∈ R.
• Multiplication distributes over addition: a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c) for all a, b, c ∈ R.

If multiplication is also commutative (a × b = b × a), it's a commutative ring. If it has a multiplicative identity (an element 1 such that 1 × a = a × 1 = a), it's a ring with unity.

Example: The set of integers ℤ with standard addition and multiplication (ℤ, +, ×) is a commutative ring with unity. (ℤ, +) is an Abelian group. Multiplication is associative (e.g., (2 × 3) × 4 = 2 × (3 × 4)), commutative (e.g., 2 × 3 = 3 × 2), and distributes over addition (e.g., 2 × (3 + 4) = 2 × 3 + 2 × 4). The multiplicative identity is 1.

1.3 Fields

A field is a special type of ring where every non-zero element has a multiplicative inverse. It's essentially a set where you can add, subtract, multiply, and divide (except by zero), and the operations behave similarly to those of rational or real numbers.

Definition of a Field: A commutative ring (F, +, ×) with unity (1 ≠ 0) is called a field if every non-zero element a ∈ F has a multiplicative inverse a^-1 ∈ F such that a × a^-1 = a^-1 × a = 1.

Example: The set of real numbers ℝ with standard addition and multiplication (ℝ, +, ×) is a field. All real numbers form an Abelian group under addition. All non-zero real numbers form an Abelian group under multiplication (e.g., for any non-zero 'a', 1/a is its inverse). Multiplication distributes over addition, and it is a commutative ring with unity (1). The set of rational numbers ℚ is also a field.

2. Linear Algebra: Vector Spaces and Related Concepts

2.1 Vector Spaces

A vector space is a fundamental structure in linear algebra, consisting of a set of vectors that can be added together and scaled (multiplied by numbers, called scalars). These operations must satisfy certain axioms to ensure they behave predictably.

Definition of a Vector Space: A vector space V over a field F is a set V equipped with two operations: vector addition (+: V × V → V) and scalar multiplication (·: F × V → V), satisfying the following 10 axioms for all u, v, w ∈ V and all a, b ∈ F:

• Vector Addition Axioms:
  1. u + v ∈ V (Closure under addition)
  2. u + v = v + u (Commutativity of addition)
  3. (u + v) + w = u + (v + w) (Associativity of addition)
  4. There exists an element 0 ∈ V such that u + 0 = u (Existence of zero vector)
  5. For each u ∈ V, there exists an element -u ∈ V such that u + (-u) = 0 (Existence of additive inverse)
• Scalar Multiplication Axioms:
  6. a · u ∈ V (Closure under scalar multiplication)
  7. a · (u + v) = a · u + a · v (Distributivity of scalar over vector addition)
  8. (a + b) · u = a · u + b · u (Distributivity of scalar over scalar addition)
  9. (a · b) · u = a · (b · u) (Associativity of scalar multiplication)
  10. 1 · u = u (Multiplicative identity for scalars)

The elements of V are called vectors, and the elements of F are called scalars. The field F is often ℝ (real numbers) or ℂ (complex numbers).

Example: The set ℝⁿ (all n-tuples of real numbers) with standard component-wise addition and scalar multiplication is a vector space over the field of real numbers ℝ. For instance, ℝ² consists of vectors like (x, y) where x, y ∈ ℝ.

2.2 Subspaces

A subspace is a subset of a vector space that is itself a vector space under the same operations as the original space. It's a