🌟 Level 3 - Topic 13: Vector Spaces (Part 1) - Introduction 🚀🌌

1) Introduction to Vector Spaces - Stepping into Abstraction 🌌

Welcome to a new realm in mathematics: Vector Spaces! So far, we've worked extensively with matrices and vectors, especially in the context of coordinate systems like \( \mathbb{R}^2 \) (2D plane) and \( \mathbb{R}^3 \) (3D space). But the concept of a 'vector space' is much broader and more abstract. It generalizes the familiar ideas of vectors in a plane or space to a vast collection of mathematical objects that share similar properties.

In essence, a vector space is a set of objects (which we will call 'vectors') where we can perform two basic operations: addition of vectors and scalar multiplication. These operations must follow certain rules, called axioms, that ensure the space behaves in a 'vector-like' way. Think of it as defining the fundamental rules of arithmetic for a generalized kind of 'vector' object.

Vector spaces are foundational in many areas of mathematics, physics, computer science, and engineering. They provide a powerful framework for dealing with linearity and transformations, which are ubiquitous in these fields. In this first part, we will explore the basic definition and examples of vector spaces to build our intuition.

2) Defining Vector Spaces: The Axioms 📜

To be a vector space, a set \(V\) must satisfy a set of rules, known as axioms, under the operations of vector addition and scalar multiplication. Let's outline these crucial axioms. For all vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) in \(V\), and scalars \( c, d \) from a field (typically real numbers \( \mathbb{R} \) in our context), the following must hold:

These ten axioms might seem abstract, but they formalize the familiar properties of vectors we've been using. Any set that satisfies these axioms is a vector space, regardless of what the 'vectors' actually are! Let's explore some examples to make this more concrete.

3) Examples of Vector Spaces 🌟

Vector spaces are much more diverse than just arrows in \( \mathbb{R}^2 \) or \( \mathbb{R}^3 \). Let's look at some key examples:

3.1) Euclidean Space \( \mathbb{R}^n \)

The most common example is the \(n\)-dimensional Euclidean space, denoted as \( \mathbb{R}^n \). For example:

• \( \mathbb{R}^2 \): The 2D plane, where vectors are ordered pairs \( (x, y) \).
• \( \mathbb{R}^3 \): The 3D space, where vectors are ordered triples \( (x, y, z) \).
• In general, \( \mathbb{R}^n \) consists of n-tuples \( (x_1, x_2, ..., x_n) \) of real numbers.

In \( \mathbb{R}^n \), vector addition and scalar multiplication are defined component-wise, and all ten axioms are satisfied. So, \( \mathbb{R}^n \) for any positive integer \(n\) is a vector space.

Example 1: Vectors in \( \mathbb{R}^2 \)

Let \( \mathbf{u} = (1, 2) \) and \( \mathbf{v} = (3, -1) \) be vectors in \( \mathbb{R}^2 \), and let \( c = 2 \) be a scalar.

Vector Addition: \( \mathbf{u} + \mathbf{v} = (1+3, 2+(-1)) = (4, 1) \in \mathbb{R}^2 \)

Scalar Multiplication: \( c\mathbf{u} = 2(1, 2) = (2\times 1, 2\times 2) = (2, 4) \in \mathbb{R}^2 \)

As you can see, the results are still within \( \mathbb{R}^2 \). You can verify that all ten axioms hold for \( \mathbb{R}^2 \) (and \( \mathbb{R}^n \) in general).

3.2) Matrix Spaces \( M_{m \times n} \)

The set of all \(m \times n\) matrices with real entries, denoted as \( M_{m \times n} \), is also a vector space. Here, the 'vectors' are matrices themselves! Vector addition is matrix addition, and scalar multiplication is scalar multiplication of a matrix, as we learned in previous topics.

Example 2: Matrices as Vectors in \( M_{2 \times 2} \)

Let \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) and \( B = \begin{bmatrix} 0 & -1 \\ 2 & 1 \end{bmatrix} \) be matrices in \( M_{2 \times 2} \), and let \( c = -1 \) be a scalar.

Matrix Addition: \( A + B = \begin{bmatrix} 1+0 & 2+(-1) \\ 3+2 & 4+1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 5 & 5 \end{bmatrix} \in M_{2 \times 2} \)

Scalar Multiplication: \( cA = (-1) \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix} \in M_{2 \times 2} \)

Again, the results remain in \( M_{2 \times 2} \). Matrix addition and scalar multiplication satisfy all vector space axioms, making \( M_{m \times n} \) a vector space.

3.3) Function Spaces \( F(\mathbb{R}) \)

Consider the set of all real-valued functions defined on the real numbers, \( F(\mathbb{R}) \). We can define vector addition as pointwise addition of functions: \( (f+g)(x) = f(x) + g(x) \), and scalar multiplication as \( (cf)(x) = c \cdot f(x) \). For example, if \( f(x) = x^2 \) and \( g(x) = \sin(x) \), then \( (f+g)(x) = x^2 + \sin(x) \), and \( (2f)(x) = 2x^2 \). With these operations, \( F(\mathbb{R}) \) is a vector space! Vectors can be functions!

3.4) Polynomial Spaces \( P_n \)

The set of all polynomials of degree at most \( n \), denoted as \( P_n \), is another important vector space. For example, \( P_2 \) is the set of all polynomials of the form \( a_2x^2 + a_1x + a_0 \), where \( a_2, a_1, a_0 \) are real numbers. Polynomial addition and scalar multiplication are defined in the usual way, and they satisfy all vector space axioms.

These examples show the versatility of the concept of a vector space. Vectors are not just arrows; they can be lists of numbers, matrices, functions, polynomials, and much more, as long as they adhere to the vector space axioms!

4) Subspaces: Vector Spaces within Vector Spaces 🏘️

Just like we have subsets of sets, we can have 'subspaces' within vector spaces. A subspace is essentially a vector space that is contained within a larger vector space, using the same operations of addition and scalar multiplication.

4.1 Definition of a Subspace

To check if a subset \(H\) of a vector space \(V\) is a subspace, we need to verify only three conditions (which ensure all ten vector space axioms are satisfied):

1. \(H\) is non-empty: The zero vector \( \mathbf{0} \) of \(V\) is in \(H\). (This ensures there's at least something in \(H\)).
2. Closure under Addition: For every \( \mathbf{u}, \mathbf{v} \in H \), \( \mathbf{u} + \mathbf{v} \) is in \(H\).
3. Closure under Scalar Multiplication: For every \( \mathbf{u} \in H \) and scalar \( c \), \( c\mathbf{u} \) is in \(H\).

If these three conditions are met, \(H\) is a subspace of \(V\).

Example 3: Subspace of \( \mathbb{R}^2 \)

Let \(H\) be the set of all vectors in \( \mathbb{R}^2 \) of the form \( \begin{bmatrix} x \\ 0 \end{bmatrix} \). Is \(H\) a subspace of \( \mathbb{R}^2 \)?

1. Non-empty? Yes, the zero vector \( \begin{bmatrix} 0 \\ 0 \end{bmatrix} \) is in \(H\) (set \(x=0\)).
2. Closure under Addition? Let \( \mathbf{u} = \begin{bmatrix} x_1 \\ 0 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} x_2 \\ 0 \end{bmatrix} \) be in \(H\). Then \( \mathbf{u} + \mathbf{v} = \begin{bmatrix} x_1 + x_2 \\ 0 \end{bmatrix} \). Since the second component is still 0, \( \mathbf{u} + \mathbf{v} \) is in \(H\).
3. Closure under Scalar Multiplication? Let \( \mathbf{u} = \begin{bmatrix} x \\ 0 \end{bmatrix} \) be in \(H\) and \( c \) be a scalar. Then \( c\mathbf{u} = \begin{bmatrix} cx \\ 0 \end{bmatrix} \). Since the second component is still 0, \( c\mathbf{u} \) is in \(H\).

Yes, \(H\) is a subspace of \( \mathbb{R}^2 \). Geometrically, \(H\) is the x-axis in the 2D plane.

Example 4: Not a Subspace of \( \mathbb{R}^2 \)

Let \(K\) be the set of all vectors in \( \mathbb{R}^2 \) of the form \( \begin{bmatrix} x \\ 1 \end{bmatrix} \). Is \(K\) a subspace of \( \mathbb{R}^2 \)?

1. Non-empty? No, the zero vector \( \begin{bmatrix} 0 \\ 0 \end{bmatrix} \) is NOT in \(K\) because the second component must be 1.

Since the first condition fails, \(K\) is not a subspace of \( \mathbb{R}^2 \). (We don't need to check the other conditions once one fails). Geometrically, \(K\) is a line parallel to the x-axis but shifted up by 1 unit; it does not pass through the origin.

Example 5: Subspace of Polynomial Space \( P_2 \)

Let \(S\) be the set of all polynomials in \( P_2 \) such that \( p(0) = 0 \). Is \(S\) a subspace of \( P_2 \)?

1. Non-empty? Yes, the zero polynomial \( p(x) = 0 \) is in \(S \) because \( p(0) = 0 \).
2. Closure under Addition? Let \( p(x), q(x) \in S \), so \( p(0) = 0 \) and \( q(0) = 0 \). Consider \( (p+q)(x) = p(x) + q(x) \). Then \( (p+q)(0) = p(0) + q(0) = 0 + 0 = 0 \). So, \( (p+q)(x) \in S \).
3. Closure under Scalar Multiplication? Let \( p(x) \in S \) (so \( p(0) = 0 \)) and \( c \) be a scalar. Consider \( (cp)(x) = c \cdot p(x) \). Then \( (cp)(0) = c \cdot p(0) = c \cdot 0 = 0 \). So, \( (cp)(x) \in S \).

Yes, \(S\) is a subspace of \( P_2 \). \(S\) consists of polynomials in \(P_2\) that have no constant term (i.e., of the form \( a_2x^2 + a_1x \)).

5) Practice Questions 🎯

6) Summary 🎉

• Vector Space Definition: A set with vector addition and scalar multiplication operations satisfying ten specific axioms (closure, associativity, commutativity, identity, inverse for addition; closure, distributivity, associativity, identity for scalar multiplication).
• Examples of Vector Spaces: \( \mathbb{R}^n \), \( M_{m \times n} \), function spaces \( F(\mathbb{R}) \), polynomial spaces \( P_n \).
• Subspace Definition: A subset \(H\) of a vector space \(V\) that is itself a vector space under the same operations.
• Subspace Test: To check if \(H\) is a subspace of \(V\), verify: 1) \( \mathbf{0} \in H \), 2) Closure under vector addition in \(H\), 3) Closure under scalar multiplication in \(H\).

Congratulations on taking your first steps into the abstract world of Vector Spaces! You've learned the fundamental definition, explored various examples, and understood the concept of subspaces. Vector spaces are a cornerstone of linear algebra, and understanding them opens doors to more advanced topics and applications. Keep exploring and practicing – the universe of linear algebra is vast and fascinating! 🚀🌌