🟣 Vector Spaces: Introduction & Axioms

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.

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.

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

• $\mathbb{R}^2$: The 2D plane, vectors are $(x, y)$.
• $\mathbb{R}^3$: The 3D space, vectors are $(x, y, z)$.
• $\mathbb{R}^n$: n-tuples $(x_1, x_2, \ldots, x_n)$.

In $\mathbb{R}^n$, vector addition and scalar multiplication are defined component-wise, and all ten axioms are satisfied.

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

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

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

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

The results are still within $\mathbb{R}^2$, satisfying closure.

The set of all $m \times n$ matrices with real entries, $M_{m \times n}$, is also a vector space. Here, the 'vectors' are matrices themselves!

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

Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -1 \\ 2 & 1 \end{bmatrix}$. Scalar $c = -1$.

Addition: $A + B = \begin{bmatrix} 1 & 1 \\ 5 & 5 \end{bmatrix} \in M_{2 \times 2}$

Scalar Mult: $cA = \begin{bmatrix} -1 & -2 \\ -3 & -4 \end{bmatrix} \in M_{2 \times 2}$

Matrix operations satisfy all vector space axioms.

Consider the set of all real-valued functions defined on the real numbers, $F(\mathbb{R})$. We define:

• Addition: $(f+g)(x) = f(x) + g(x)$
• Scalar Mult: $(cf)(x) = c \cdot f(x)$

For example, if $f(x) = x^2$ and $g(x) = \sin(x)$, then $(f+g)(x) = x^2 + \sin(x)$. This set forms a vector space where the "vectors" are functions!

The set of all polynomials of degree at most $n$, denoted as $P_n$, is a vector space. For example, $P_2$ contains polynomials like $a_2x^2 + a_1x + a_0$.

Example 3: Subspace of $\mathbb{R}^2$ (Line through Origin)

Let $H$ be the set of vectors $\begin{bmatrix} x \\ 0 \end{bmatrix}$. Is $H$ a subspace?

1. Zero vector? Yes, $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$ is in $H$ (when $x=0$).
2. Addition? $\begin{bmatrix} x_1 \\ 0 \end{bmatrix} + \begin{bmatrix} x_2 \\ 0 \end{bmatrix} = \begin{bmatrix} x_1+x_2 \\ 0 \end{bmatrix}$. Still in $H$.
3. Scalar Mult? $c\begin{bmatrix} x \\ 0 \end{bmatrix} = \begin{bmatrix} cx \\ 0 \end{bmatrix}$. Still in $H$.

Conclusion: Yes, $H$ (the x-axis) is a subspace.

Example 4: Not a Subspace (Shifted Line)

Let $K$ be vectors $\begin{bmatrix} x \\ 1 \end{bmatrix}$. Is $K$ a subspace?

Check Zero Vector: Is $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$ in $K$? No, because the second component must be 1, not 0.

Conclusion: $K$ is not a subspace. Geometrically, it's a line that doesn't pass through the origin.

Example 5: Subspace of Polynomials

Let $S$ be polynomials in $P_2$ where $p(0) = 0$. Is $S$ a subspace?

• Zero? $p(x)=0$ implies $p(0)=0$. Yes.
• Addition? If $p(0)=0, q(0)=0$, then $(p+q)(0) = 0+0=0$. Yes.
• Scalar? If $p(0)=0$, then $(cp)(0) = c(0) = 0$. Yes.

Conclusion: Yes, $S$ is a subspace.

1️⃣ Signal Processing

Signals (audio, radio) are treated as vectors in infinite-dimensional function spaces. Operations like filtering are linear transformations on these spaces.

2️⃣ Quantum Mechanics

The state of a quantum system is described by a vector in a complex vector space called a Hilbert space. Superposition is simply vector addition!

3️⃣ Computer Graphics & Color

RGB colors form a vector space. Adding colors (light) is vector addition. Scaling a color vector changes its brightness.

4️⃣ Data Science & AI

Data points are represented as "feature vectors" in high-dimensional spaces. Machine learning algorithms find patterns (like planes) separating these vectors.