Vector

The Everyday Idea of a Vector

Back when we were at School, our Physics teacher have told us the general definition of a vector: Vector is a quantity that have both Magnitude & Direction. Most of the cases, the definition stands tall: a vector is something that has both Magnitude (size or length) and Direction.

• "Walk 5 blocks (Magnitude) North-East (Direction)."

This is the standard, geometric way to think about a vector - as an arrow pointing from one place to another.

In linear algebra and data science, we often use a different, but equivalent, representation. We think of a vector as an ordered list of numbers, like this:

$$u = \begin{pmatrix} u_1 \\ u_2 \\ \vdots \\ u_n \end{pmatrix}$$

• This is an n-component vector. The numbers $u_1, u_2, \dots, u_n$ are the components. Components are also called as Dimensions.
• It is also called an ordered n-tuple. "Ordered" means the sequence matters ($\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ is different from $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$).

Example: A 2-Component Vector

Let's look at the simplest case: a vector with 2 components.

• 2-component vector: $u = \begin{pmatrix} u_1 \\ u_2 \end{pmatrix}$.
• Specific example: $u = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$.

This vector can be plotted on a 2D graph with axes $x_1$ and $x_2$. Let's plot:

The vector starts at the origin (0, 0) and ends at the point (2, 3). The arrow itself is what represents the vector.

We are able to plot this in a 2D graph, because the vector have two components. We could also possibly represent a 3D vector but no more than that (We are talking in the sense of a 2D graph). Also this is the case for the current transformer architecture. Each word is represented as a multi-dimensional vector. Plotting it will be a havoc.

Real vs. Abstract Vectors

The vector $u = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ is a Real vector because both of its components (2 and 3) are real numbers.

Throughout the series, we will primarily deal with real vectors, but remember that components could theoretically come from other mathematical systems, like complex numbers!

Ex: Biological Parameters Example:

Now, let's step away from geometry. Imagine you are tracking a patient's health. You measure 5 different parameters. You can put all these measurements into a single 5-component vector:

$$u = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \end{pmatrix}$$

Represented as a Table:

In this context, the vector isn't a geometric arrow, but a single, organized collection of 5 related data points. This abstraction is the foundation of modern data analysis!

Let's say we are collecting 100 patient's data.

• Collect 5 parameters ($u_1, u_2, u_3, u_4, u_5$) from 100 different patients ($P_1, \dots, P_{100}$).
• This data can be represented as a matrix.
• The matrix will have dimensions $100 \times 5$.
  • Rows represent the patients.
  • Columns represent the different parameters we measure.

Then the Data Representation:

$$ \begin{pmatrix} u_1^{(1)} & u_2^{(1)} & u_3^{(1)} & u_4^{(1)} & u_5^{(1)} \\ u_1^{(2)} & u_2^{(2)} & u_3^{(2)} & u_4^{(2)} & u_5^{(2)} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ u_1^{(100)} & u_2^{(100)} & u_3^{(100)} & u_4^{(100)} & u_5^{(100)}\\ \end{pmatrix} $$

This we can say that, a vector can also be a powerful way to organize data.

Field $\mathbb{F}$

Theory:

• A Field 𝔽 is a non-empty collection of elements with two operations:
  • Field addition (+)
  • Field multiplication (⋅).
• These operations must satisfy the following properties:

Multiplicative Inverse

• $a^{-1}$ is the multiplicative inverse of $a$.

Example 1: Set of Real Numbers, $\mathbb{R}$

Consider the set of real numbers, $\mathbb{R}$. Let's check if it's a field with the standard operations of addition (+) and multiplication (⋅).

• (i) $0 \in \mathbb{R}$. (Additive identity exists)
• (ii) For any two real numbers $a, b \in \mathbb{R}, a+b \in \mathbb{R}$. (Closure under addition)
• (iii) For every $a \in \mathbb{R}, (-a) \in \mathbb{R}$ such that $a + (-a) = -a + a = 0$. (Additive inverse exists)
• (iv) $1 \in \mathbb{R}$. (Multiplicative identity exists)
• (v) For any two real numbers $a, b \in \mathbb{R}, a \cdot b \in \mathbb{R}$. (Closure under multiplication)
• (vi) For every $a \neq 0, a \in \mathbb{R}$, there exists an $a^{-1} \in \mathbb{R}$ such that $a^{-1} \cdot a = a \cdot a^{-1} = 1$. (Multiplicative inverse exists for non-zero elements)

Conclusion: $\mathbb{R}$ is a field with operations $+$ and $\cdot$.

Example 2: Set of Integers, $\mathbb{N}$

Consider the set of integers, $\mathbb{N}$. Let's check the field properties.

• (i) $0 \in \mathbb{N}$.
• (ii) $a+b \in \mathbb{N}$ for $a, b \in \mathbb{N}$.
• (iii) For every $a \in \mathbb{N}, (-a) \in \mathbb{N}$ such that $a+(-a) = 0$. (This is true)
• (iv) $1 \in \mathbb{N}$.
• (v) $a \cdot b \in \mathbb{N}$ for any two integers $a, b$.
• (vi) Suppose $a \in \mathbb{N}$ (e.g., $a=2$). Then the multiplicative inverse $a^{-1} = 1/2 \notin \mathbb{N}$.

Conclusion: The set of integers is not a field.

Example 3: Integers Modulo 5, $\mathbb{Z}_5$

Consider the set of integers modulo 5. This is the set of remainders when an integer is divided by 5.

• $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$.
• The operations are:
  • $\oplus_5$: Add modulo 5.
  • $\otimes_5$: Multiply modulo 5.

Integers Modulo 5, $\mathbb{Z}_5$ (cont'd)

• The set $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$ is a field with operations $\oplus_5$ and $\otimes_5$.

Properties of $\mathbb{Z}_5$

• Addition Modulo 5 ($\oplus_5$):

  

  • The table for addition modulo 5 shows that the set is closed under addition mod 5.
  • Additive Identity (0): The first row and column show that adding 0 to any element results in the same element.
  • Additive Inverse: For every element $a \in \mathbb{Z}_5$, the additive inverse $(-a)$ exists in $\mathbb{Z}_5$. The table shows this:
    • $0$'s inverse is $0 (0+0=0)$
    • $1$'s inverse is $4$ ($1+4=0$)
    • $2$'s inverse is $3 (2+3=0)$
    • $3$'s inverse is $2 (3+2=0)$
    • $4$'s inverse is $1$ ($4+1=0$)

• Multiplication Modulo 5 ($\otimes_5$):

  

  • The table for multiplication modulo 5 shows that the set is closed under multiplication mod 5.
  • (iv) Multiplicative Identity: The element $1 \in \mathbb{Z}_5$ is the multiplicative identity.
  • (v) Closure: For any $a, b \in \mathbb{Z}_5$, $a \otimes_5 b \in \mathbb{Z}_5$.
  • (vi) Multiplicative Inverse: For every non-zero element $a \in \mathbb{Z}_5$, there exists a unique integer $a^{-1} \in \mathbb{Z}_5$ such that $a \otimes_5 a^{-1} \equiv 1 \pmod{5}$.
    • $a=1 \implies a^{-1}=1$ ($1 \otimes_5 1 = 1$)
    • $a=2 \implies a^{-1}=3$ ($2 \otimes_5 3 = 6 \equiv 1 \pmod{5}$)
    • $a=3 \implies a^{-1}=2$ ($3 \otimes_5 2 = 6 \equiv 1 \pmod{5}$)
    • $a=4 \implies a^{-1}=4$ ($4 \otimes_5 4 = 16 \equiv 1 \pmod{5}$)

Conclusion: The set $\mathbb{Z}_5$ is a field with the operations of addition and multiplication modulo 5.