Vector space & Subspace

2.1 Introduction to Vector Spaces

• Vector Space: Collection of objects called vectors, manipulated using addition and scalar multiplication.

• Importance: Plays a vital role in sciences, engineering, and particularly in machine learning.

  • Data points represented as vectors; each dimension corresponds to a specific feature.

• Vectors in R: Defined by magnitude and direction.

  • Geometrically represented as line segments in a plane.

  • Notation: A vector in R^2 is represented by V = (u1, u2).

• Generalization:

  • Vectors in

    • 1-space (R) are real numbers.

    • 2-space (R^2) are ordered pairs.

    • 3-space (R^3) are ordered triplets.

    • n-space (R^n) are n-tuples.

2.2 Operations on Vectors

• Addition of Vectors:

  • If u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), then:

    • Addition: u + v = (u1 + v1, u2 + v2, ..., un + vn).

• Scalar Multiplication:

  • For a scalar a:

    • a.u = (au1, au2, ..., a*un).

  • Example: If v is multiplied by 2, then the resulting vector is twice the length of the original vector, retaining its direction.

2.3 Vector Space Definition

Definition 2.1.1: Vector Space

• A set V with operations of addition (+) and scalar multiplication (•) is a vector space over real numbers R if:

  1. Closure under addition: u + v ∈ V for all u, v ∈ V.

  2. Commutative Law: u + v = v + u.

  3. Associative Law: u + (v + w) = (u + v) + w.

  4. Identity Element: There exists a zero vector 0 ∈ V such that u + 0 = u.

  5. Additive Inverse: For every u ∈ V, there exists an -u ∈ V such that u + (-u) = 0.

  6. Closure under scalar multiplication: a.u ∈ V for all a ∈ R, and u ∈ V.

  7. Associative Property of Scalar Multiplication: a(b.u) = (ab).u.

  8. Identity Element for Scalar Multiplication: 1.u = u.

  9. Distributive Property: a(u + v) = a.u + a.v.

  10. Distributive Property for Scalars: (a + b).u = a.u + b.u.

2.4 Example of Vector Space R^n

• Set of all n-tuples of real numbers: R^n = {(u1, u2, ..., un) : ui ∈ R, for i = 1, 2,... , n}.

• Operations:

  • Vector Addition: u + v = (u1 + v1, u2 + v2, ..., un + vn).

  • Scalar Multiplication:

    • a.v = (a.u1, a.u2, ..., a.un).

• Confirming vector space properties for R^n under given operations:

  • Closure under addition and scalar multiplication verified.

  • Associative, Commutative and Identity properties established.

2.5 Understanding Matrix Spaces

• 2 x 3 matrices over R:

  • Addition is defined as: A + B = (a11 + b11, a12 + b12, ...).

  • Scalar multiplication defined as: c.A = (ca11, ca12, ...).

• Matrix addition and scalar multiplication show closure, ensuring it forms a vector space.

2.6 Properties Summary

• Vector Space Examples: Polynomials, continuous functions, etc.

• Theorems of Scalar Multiplication:

  1. 0.v = 0

  2. c.0 = 0

  3. If c.v = 0, then either c = 0 or v = 0.

  4. (-1).v = -v.

• Additional review of certain subsets and their properties in confirming or denying vector space status.

2.1 Introduction to Vector Spaces

Vector Space: A vector space is a collection of objects called vectors that can be manipulated through two primary operations: addition and scalar multiplication. These operations must satisfy certain criteria and properties to qualify the set as a vector space.

Importance of Vector Spaces: Vector spaces are foundational to various fields in mathematics and play a vital role in sciences, engineering, and particularly in machine learning. In these fields, data points are often represented as vectors, where each dimension corresponds to a specific feature of the data.

Vectors in R: In mathematics, vectors are defined by their magnitude (size) and direction. Geometrically, vectors can be represented as line segments in a coordinate plane, allowing for visualization and understanding of their properties. The notation used for a vector in R^2 (two-dimensional space) is represented as V = (u1, u2), where u1 and u2 are the components of the vector.

Generalization of Vectors

Vectors can be generalized to different dimensional spaces:

• 1-space (R): Represents real numbers, indicating a single dimension.

• 2-space (R^2): Represents ordered pairs, visualized in a two-dimensional plane.

• 3-space (R^3): Represents ordered triplets, utilized in three-dimensional contexts. Commonly used to model physical spaces.

• n-space (R^n): General form for any n-dimensional space, represented by n-tuples of real numbers.

2.2 Operations on Vectors

Addition of Vectors:

If u = (u1, u2, ..., un) and v = (v1, v2, ..., vn), then:

• Vector Addition: Defined as u + v = (u1 + v1, u2 + v2, ..., un + vn). This operation combines two vectors to produce a new vector.

Scalar Multiplication:

A scalar is a real number, denoted as a. The scalar multiplication of vector u by a scalar a yields:

• Scalar Multiplication: a.u = (au1, au2, ..., a.un). This operation scales the length of the vector while retaining its direction. For instance, if vector v is multiplied by 2, the resulting vector is twice the length of the original vector, directed in the same direction.

2.3 Vector Space Definition

Definition 2.1.1: Vector Space

A set V with operations of addition (+) and scalar multiplication (•) is a vector space over the real numbers R if it satisfies the following properties:

1. Closure under Addition: For all u, v ∈ V, the result of u + v must also be in V.

2. Commutative Law: The order of vector addition does not affect the outcome: u + v = v + u.

3. Associative Law: Grouping of vectors does not affect the sum: u + (v + w) = (u + v) + w.

4. Identity Element: There exists a zero vector 0 ∈ V such that for every vector u in V, u + 0 = u.

5. Additive Inverse: For each vector u in V, there exists an additive inverse -u such that u + (-u) = 0.

6. Closure under Scalar Multiplication: For any scalar a ∈ R and any vector u ∈ V, the product a.u ∈ V.

7. Associative Property of Scalar Multiplication: a(b.u) = (ab).u, meaning scalar multiplication is associative.

8. Identity Element for Scalar Multiplication: 1.u = u. The scalar 1 acts as an identity element.

9. Distributive Property: For scalars a, b and vector u, a(u + v) = a.u + a.v holds true.

10. Distributive Property for Scalars: (a + b).u = a.u + b.u for scalars a and b.

2.4 Example of Vector Space R^n

The set of all n-tuples of real numbers is defined as follows:

• R^n: R^n = {(u1, u2, ..., un) : ui ∈ R for i = 1, 2,... , n}. This set encompasses all possible combinations of n numbers from the real number line.

Operations in R^n:

1. Vector Addition: u + v = (u1 + v1, u2 + v2, ..., un + vn).

2. Scalar Multiplication: a.v = (a.u1, a.u2, ..., a.un).

This confirms vector space properties for R^n under these operations, including closure under addition and scalar multiplication, as well as the verification of associative, commutative, and identity properties.

2.5 Understanding Matrix Spaces

Consider 2 x 3 matrices over R:

• Matrix Addition: Defined as A + B = (a11 + b11, a12 + b12, ...), where A and B are 2 x 3 matrices.

• Scalar Multiplication: Defined as c.A = (ca11, ca12, ...), scaling each element in the matrix by the scalar c.

These operations demonstrate closure under both addition and scalar multiplication, indicating that a set of matrices forms a vector space.

2.6 Properties Summary

Vector Space Examples: Common examples of vector spaces include polynomial functions, continuous functions, and more.

Theorems of Scalar Multiplication:

1. Zero Multiplication: 0.v = 0, meaning scaling any vector by zero results in the zero vector.

2. Identity of Zero: c.0 = 0, reflecting that multiplying the zero vector by any scalar results in the zero vector.

3. Non-Trivial Scalar Multiplication: If c.v = 0, it implies either c = 0 or v = 0.

4. Negation: (-1).v = -v shows that multiplying a vector by -1 yields its additive inverse.

Additional Review:

A thorough examination of various subsets and their properties is necessary to confirm or deny their status as vector spaces based on the previously defined criteria and properties.

2.1 Introduction to Vector Spaces

Vector Space: A vector space is a collection of objects called vectors that can be manipulated through addition and scalar multiplication.

Importance: Vital in sciences, engineering, and machine learning, with data points represented as vectors.

Types of Vectors:

• 1-space (R): Real numbers.

• 2-space (R²): Ordered pairs.

• 3-space (R³): Ordered triplets.

• n-space (Rⁿ): n-tuples of real numbers.

2.2 Operations on Vectors

• Addition: If u = (u1, u2) and v = (v1, v2), then u + v = (u1 + v1, u2 + v2).

  Example Question: If u = (2, 3) and v = (4, 5), what is u + v?Answer: u + v = (6, 8)

• Scalar Multiplication: For a scalar a, a.u = (au1, au2).

  Example Question: If u = (3, 2) and a = 4, what is a.u?Answer: a.u = (12, 8)

2.3 Vector Space Properties

A set V is a vector space if it satisfies the following properties:

1. Closure under Addition and Scalar Multiplication.

2. Commutative and Associative Laws.

3. Identity Elements and Additive Inverses.

Example Question: Is R² a vector space? Why?Answer: Yes, R² is a vector space because it satisfies these properties.

2.4 Example: Rⁿ

Rⁿ is defined as all n-tuples of real numbers. Operations confirm closure and properties.

Example Question: If u = (1, 2) and v = (3, 4), what is u + v?Answer: u + v = (4, 6)

2.5 Matrix Spaces

For 2x3 matrices:

• Addition: A + B.

• Scalar Multiplication: c.A.

  Example Question: If A = [[1, 2, 3], [4, 5, 6]] and B = [[7, 8, 9], [10, 11, 12]], what is A + B?Answer: A + B = [[8, 10, 12], [14, 16, 18]]

2.6 Properties Summary

• Examples: Polynomials, continuous functions.

• Theorems: 0.v = 0, (-1).v = -v, and others.