Chapter 2: Fundamental Theory of Vector Spaces

In-Depth Understanding of Vectors

Mathematical Definition of Vectors

A vector is an ordered array, typically represented as: $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$

where $v_i$ is the $i$-th component of the vector.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

# Examples of vectors in different dimensions
v_1d = np.array([5])                    # 1D vector
v_2d = np.array([3, 4])                 # 2D vector
v_3d = np.array([1, 2, 3])              # 3D vector
v_nd = np.array([1, -2, 0, 5, 3, -1])   # High-dimensional vector

print(f"1D vector: {v_1d}")
print(f"2D vector: {v_2d}")
print(f"3D vector: {v_3d}")
print(f"6D vector: {v_nd}")

Geometric Representation of Vectors

# Geometric representation of 2D vectors
fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# Position vectors
v1 = np.array([3, 2])
v2 = np.array([-2, 3])
v3 = np.array([1, -2])

# First subplot: multiple vectors
axes[0].quiver(0, 0, v1[0], v1[1], angles='xy', scale_units='xy', scale=1,
               color='red', width=0.005, label='v1=(3,2)')
axes[0].quiver(0, 0, v2[0], v2[1], angles='xy', scale_units='xy', scale=1,
               color='blue', width=0.005, label='v2=(-2,3)')
axes[0].quiver(0, 0, v3[0], v3[1], angles='xy', scale_units='xy', scale=1,
               color='green', width=0.005, label='v3=(1,-2)')

axes[0].set_xlim(-3, 4)
axes[0].set_ylim(-3, 4)
axes[0].grid(True, alpha=0.3)
axes[0].legend()
axes[0].set_title('Vectors in Different Directions')
axes[0].set_xlabel('x')
axes[0].set_ylabel('y')

# Second subplot: vector magnitude
v = np.array([4, 3])
magnitude = np.linalg.norm(v)

axes[1].quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1,
               color='purple', width=0.008, label=f'v=({v[0]},{v[1]})')
axes[1].plot([0, v[0]], [0, v[1]], 'purple', linewidth=2)
axes[1].text(v[0]/2, v[1]/2 + 0.3, f'|v| = {magnitude:.2f}',
             fontsize=12, ha='center')

axes[1].set_xlim(-1, 5)
axes[1].set_ylim(-1, 4)
axes[1].grid(True, alpha=0.3)
axes[1].legend()
axes[1].set_title('Vector Magnitude')
axes[1].set_xlabel('x')
axes[1].set_ylabel('y')

plt.tight_layout()
plt.show()

Basic Vector Operations

Vector Addition

The addition of two vectors is defined as the addition of corresponding components: $\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n + v_n \end{bmatrix}$

# Vector addition example
u = np.array([2, 1])
v = np.array([1, 3])
result = u + v

print(f"Vector u: {u}")
print(f"Vector v: {v}")
print(f"u + v: {result}")

# Geometric representation of vector addition (parallelogram rule)
plt.figure(figsize=(8, 6))

# Draw vector u
plt.arrow(0, 0, u[0], u[1], head_width=0.1, head_length=0.1,
          fc='red', ec='red', label='u')
# Draw vector v
plt.arrow(0, 0, v[0], v[1], head_width=0.1, head_length=0.1,
          fc='blue', ec='blue', label='v')
# Draw vector v starting from the endpoint of u
plt.arrow(u[0], u[1], v[0], v[1], head_width=0.1, head_length=0.1,
          fc='blue', ec='blue', linestyle='--', alpha=0.7)
# Draw vector u starting from the endpoint of v
plt.arrow(v[0], v[1], u[0], u[1], head_width=0.1, head_length=0.1,
          fc='red', ec='red', linestyle='--', alpha=0.7)
# Draw result vector
plt.arrow(0, 0, result[0], result[1], head_width=0.15, head_length=0.15,
          fc='green', ec='green', linewidth=2, label='u + v')

plt.grid(True, alpha=0.3)
plt.legend()
plt.xlim(-0.5, 4)
plt.ylim(-0.5, 5)
plt.title('Geometric Representation of Vector Addition')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

Scalar Multiplication

The multiplication of a scalar and a vector is defined as: $c\mathbf{v} = c\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} = \begin{bmatrix} cv_1 \\ cv_2 \\ \vdots \\ cv_n \end{bmatrix}$

# Scalar multiplication example
v = np.array([2, 1])
scalars = [-2, -1, 0.5, 1, 2]

plt.figure(figsize=(10, 6))

colors = ['red', 'orange', 'yellow', 'blue', 'green']
for i, c in enumerate(scalars):
    scaled_v = c * v
    plt.arrow(0, 0, scaled_v[0], scaled_v[1], head_width=0.1, head_length=0.1,
              fc=colors[i], ec=colors[i], label=f'{c}v')

plt.grid(True, alpha=0.3)
plt.legend()
plt.xlim(-5, 5)
plt.ylim(-3, 3)
plt.title('Geometric Effect of Scalar Multiplication')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

# Numerical computation
print("Scalar multiplication results:")
for c in scalars:
    result = c * v
    print(f"{c} * {v} = {result}")

Axiomatic Definition of Vector Spaces

A vector space (linear space) is a set $V$ whose elements are called vectors and satisfy the following axioms:

Addition Axioms

Let $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$:

1. Closure: $\mathbf{u} + \mathbf{v} \in V$
2. Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$
3. Associativity: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$
4. Zero element: There exists $\mathbf{0} \in V$ such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$
5. Negative element: For each $\mathbf{v} \in V$, there exists $-\mathbf{v} \in V$ such that $\mathbf{v} + (-\mathbf{v}) = \mathbf{0}$

Scalar Multiplication Axioms

Let $a, b$ be scalars, $\mathbf{u}, \mathbf{v} \in V$:

6. Closure: $a\mathbf{v} \in V$
7. Distributivity 1: $a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}$
8. Distributivity 2: $(a + b)\mathbf{v} = a\mathbf{v} + b\mathbf{v}$
9. Associativity: $(ab)\mathbf{v} = a(b\mathbf{v})$
10. Identity: $1\mathbf{v} = \mathbf{v}$

# Verify vector space axioms
def verify_vector_space_axioms():
    # Define some vectors
    u = np.array([1, 2, 3])
    v = np.array([4, 5, 6])
    w = np.array([7, 8, 9])
    zero = np.array([0, 0, 0])

    # Scalars
    a, b = 2, 3

    print("Verifying vector space axioms:")
    print("=" * 40)

    # Addition commutativity
    print(f"1. Commutativity: u + v = {u + v}")
    print(f"                  v + u = {v + u}")
    print(f"   Equal? {np.array_equal(u + v, v + u)}")

    # Addition associativity
    print(f"\n2. Associativity: (u + v) + w = {(u + v) + w}")
    print(f"                  u + (v + w) = {u + (v + w)}")
    print(f"   Equal? {np.array_equal((u + v) + w, u + (v + w))}")

    # Zero vector
    print(f"\n3. Zero vector: u + 0 = {u + zero}")
    print(f"                u = {u}")
    print(f"   Equal? {np.array_equal(u + zero, u)}")

    # Negative vector
    neg_u = -u
    print(f"\n4. Negative vector: u + (-u) = {u + neg_u}")
    print(f"   Is zero? {np.allclose(u + neg_u, zero)}")

    # Scalar multiplication distributivity
    print(f"\n5. Distributivity 1: a(u + v) = {a * (u + v)}")
    print(f"                     au + av = {a * u + a * v}")
    print(f"   Equal? {np.array_equal(a * (u + v), a * u + a * v)}")

    print(f"\n6. Distributivity 2: (a + b)u = {(a + b) * u}")
    print(f"                     au + bu = {a * u + b * u}")
    print(f"   Equal? {np.array_equal((a + b) * u, a * u + b * u)}")

verify_vector_space_axioms()

Examples of Vector Spaces

Common Vector Spaces

# 1. R^n space - n-dimensional real vector space
print("1. R^n space examples:")
R2_vectors = [np.array([1, 2]), np.array([3, -1]), np.array([0, 5])]
R3_vectors = [np.array([1, 2, 3]), np.array([4, 5, 6]), np.array([7, 8, 9])]

print("Vectors in R^2:", R2_vectors)
print("Vectors in R^3:", R3_vectors)

# 2. Polynomial space
print("\n2. Polynomial space P_n:")
print("Polynomials of degree at most 2: {a + bx + cx^2 | a, b, c ∈ R}")
print("Examples: 3 + 2x - x^2, 5x + 4x^2, 7 - 3x")

# 3. Matrix space
print("\n3. Matrix space M_{m×n}:")
matrix_space_example = [
    np.array([[1, 2], [3, 4]]),
    np.array([[5, 6], [7, 8]]),
    np.array([[0, 1], [-1, 0]])
]
print("Elements in 2×2 matrix space:")
for i, matrix in enumerate(matrix_space_example):
    print(f"Matrix {i+1}:\n{matrix}")

# 4. Function space
print("\n4. Function space:")
print("Continuous function space C[0,1]: all continuous functions on interval [0,1]")
print("Examples: f(x) = x^2, g(x) = sin(x), h(x) = e^x")

Subspace Theory

Definition of Subspace

A subset $W$ of a vector space $V$ is called a subspace of $V$ if $W$ is itself a vector space.

Subspace Criterion Theorem

$W$ is a subspace of $V$ if and only if:

1. $\mathbf{0} \in W$ (contains zero vector)
2. For any $\mathbf{u}, \mathbf{v} \in W$, we have $\mathbf{u} + \mathbf{v} \in W$ (closed under addition)
3. For any $c \in \mathbb{R}$, $\mathbf{v} \in W$, we have $c\mathbf{v} \in W$ (closed under scalar multiplication)

def is_subspace(vectors, test_vectors=None):
    """
    Check whether a given set of vectors spans a subspace
    """
    vectors = np.array(vectors)

    # Check if contains zero vector
    zero_vector = np.zeros(vectors.shape[1])
    contains_zero = any(np.allclose(v, zero_vector) for v in vectors)

    print(f"Contains zero vector: {contains_zero}")

    # Check closure under addition (simple test)
    if len(vectors) >= 2:
        v1, v2 = vectors[0], vectors[1]
        sum_vector = v1 + v2
        print(f"v1 + v2 = {sum_vector}")

    # Check closure under scalar multiplication
    if len(vectors) >= 1:
        v1 = vectors[0]
        scaled = 2 * v1
        print(f"2 * v1 = {scaled}")

    return contains_zero

# Example 1: Line through origin in R^3
print("Example 1: Line through origin span{(1,2,3)}")
line_vectors = [np.array([0, 0, 0]), np.array([1, 2, 3]), np.array([2, 4, 6])]
is_subspace(line_vectors)

print("\n" + "="*50)

# Example 2: Plane through origin in R^3
print("Example 2: Plane through origin span{(1,0,0), (0,1,0)}")
plane_vectors = [np.array([0, 0, 0]), np.array([1, 0, 0]),
                 np.array([0, 1, 0]), np.array([1, 1, 0])]
is_subspace(plane_vectors)

Common Types of Subspaces

graph TD
    A[Vector Space V] --> B[Trivial Subspaces]
    A --> C[Non-trivial Subspaces]

    B --> D[Zero Subspace {0}]
    B --> E[Entire Space V]

    C --> F[Line Subspace]
    C --> G[Plane Subspace]
    C --> H[Hyperplane Subspace]

    F --> I[span{v}: v ≠ 0]
    G --> J[span{u,v}: u,v linearly independent]
    H --> K[Kernel space ker(T)]

Importance of Zero and Negative Vectors

Properties of Zero Vector

The zero vector $\mathbf{0}$ is a special element in vector space:

# Demonstration of zero vector properties
zero_2d = np.array([0, 0])
zero_3d = np.array([0, 0, 0])

v = np.array([3, 4])
w = np.array([1, 2, 3])

print("Properties of zero vector:")
print(f"1. v + 0 = {v} + {zero_2d} = {v + zero_2d}")
print(f"2. 0 + v = {zero_2d} + {v} = {zero_2d + v}")
print(f"3. 0 * v = 0 * {v} = {0 * v}")
print(f"4. ||0|| = {np.linalg.norm(zero_2d)}")

# Role of zero vector in linear combinations
a, b, c = 2, -3, 5
u1 = np.array([1, 0])
u2 = np.array([0, 1])
u3 = np.array([0, 0])  # Zero vector

linear_comb = a * u1 + b * u2 + c * u3
print(f"\nLinear combination: {a}*{u1} + {b}*{u2} + {c}*{u3} = {linear_comb}")

Properties of Negative Vector

# Demonstration of negative vector properties
v = np.array([3, -2, 1])
neg_v = -v

print("Properties of negative vector:")
print(f"Vector v: {v}")
print(f"Negative vector -v: {neg_v}")
print(f"v + (-v) = {v + neg_v}")
print(f"||v|| = {np.linalg.norm(v):.3f}")
print(f"||-v|| = {np.linalg.norm(neg_v):.3f}")

# Visualize positive and negative vectors
plt.figure(figsize=(8, 6))
v_2d = np.array([3, 2])
neg_v_2d = -v_2d

plt.arrow(0, 0, v_2d[0], v_2d[1], head_width=0.2, head_length=0.2,
          fc='blue', ec='blue', label='v')
plt.arrow(0, 0, neg_v_2d[0], neg_v_2d[1], head_width=0.2, head_length=0.2,
          fc='red', ec='red', label='-v')

plt.grid(True, alpha=0.3)
plt.axis('equal')
plt.legend()
plt.xlim(-4, 4)
plt.ylim(-3, 3)
plt.title('Vector and Its Negative')
plt.xlabel('x')
plt.ylabel('y')
plt.show()

Chapter Summary

This chapter explored the fundamental theory of vector spaces in depth:

Through this chapter, we have established a rigorous foundation in vector space theory, preparing us for subsequent learning of linear dependence, basis and dimension, and other concepts.