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Google Slides:

https://docs.google.com/presentation/d/1hujfHzj6w_F8WphpoW7eXvdiMjo2DHzwChKA4J9fTsk/edit?usp=sharing

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The flow of understanding math for me looks like this: context → symbol → terminology.

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This chapter focusing on foundational concepts including vector notation, vector addition, scalar multiplication, and the formal definition of n-dimensional vector spaces.

DAY 1: INTRODUCTION TO Rⁿ

Date: November 11, 2025 11:00 AM (GMT+7)

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How to read this note: Each table row is explained below the table using the same numbering.

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Table 1: Defining Vectors as Lists of Numbers

	Goal	Step / Method	Justification / Example
1	Represent a list of values compactly	Use a single symbol with subscripts: $w₁, w₂, ..., wₙ$	Subscript denotes position in the list. Example: weights $w₁ = 156, w₂ = 125, ..., w₈ = 193$
2	Formalize the list as a mathematical object	Write as an ordered tuple: $w = (w₁, w₂, ..., wₙ)$	This tuple is called a linear array or vector
3	Distinguish vector elements from other fields	Call elements scalars (from $R$ or $C$)	Scalars are the individual numerical entries of the vector

1. Compact Representation via Indexed Notation

For the first point, however, ‘subscripts’ are not mere “position markers” or “labeling.” Subscript defines a coordinate system, which means $w₃$ means something fundamentally different from $w₁$.

Goal
Method
Example

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Why R vs C Matters?

Because it fundamentally determines the following points:

What operations are algebraically valid
The geometric structure of the vector space
Which linear transformation exist

Point 1: Algebraic Consequences

Real scalars (R):

Scaling is limited to stretching/compressing along a line
Every nonzero scalar has sign (positive/negative direction)
Polynomial equations may have no solutions (e.g., x² + 1 = 0 in R)

Complex scalars (C):

Scaling includes rotation (multiplication by $e^{i\theta}$ rotates by θ)
Every polynomial has roots
You gain access to phase information | Phase is the angular component of a complex number when written in polar form. It represents rotation.

Point 2: Geometric Consequences

Real vector spaces:

Inner product gives angles and lengths (Euclidean geometry)

Complex vector spaces:

The complex vector (1, i) has "length" √2, but if you plot it as (1,0,0,1) in R⁴, that interpretation hides the phase structure.

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2. Vectors as Ordered n-tuples in Rⁿ

A disclaimer for the second point: the notation is not a structure. Those notations written referring to a ‘tuple.’ A tuple becomes a vector only when it lives in a ‘vector space’ with defined operations (addition, scalar multiplication).