**EUCLIDEAN SPACES**

## A GEOMETRIC JOURNEY FROM DOT TO DISTANCE & METRIC; VECTOR, INNER PRODUCT & SPACES ...

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### Overview

This course provides a comprehensive exploration of Euclidean space and its key concepts, guiding students on a journey from the fundamental notions of dots and distances to more advanced topics such as metrics, vectors, inner products, and spaces. The course aims to develop a deep understanding of Euclidean space and its applications in various fields.

### course outline

Vector, Distance & Metric

- Overview on the difference between static point and vector
- Fundamental properties of distance and their geometric interpretation
- Defining the metric by generalizing distance between two vectors

1+1 and Vector Spaces

- Defining 1+1 in a collection of vectors and how -(-1) becomes 1
- Roles of these two identities on vector addition and scalar multiplication
- Vector space and their properties

Cartesian Product, Vector Addition & Coordinate System

- Defining Cartesian Product
- Construction of a new space using Cartesian Product
- Overview on how vector addition is associated with the cartesian product in coordinate representation of any vector in that space

Dimensions & Linearity

- Defining a function and geometric overview of its linear properties
- Geometric representation of linearly dependent and independent vectors
- Defining bases and dimensions

Pythagoras Theorem & Component of Vectors

- Geometric proof of Pythagoras Theorem
- Determining norm or length of a vector
- Notion of vector addition and components

Projection, Inner Product & Euclidean Spaces

- Representation of projection and generalization
- Defining inner products and geometric representation of all its properties
- Defining Euclidean Spaces

Norms & Law of Cosines

- Defining norms w.r.t inner product
- Finding out relation among norm, inner product, Pythagoras theorem and law of cosines
- Representation of inner product w.r.t coordinates

Cauchy-Schwarz Inequality & Metric Spaces

- Geometric proof of Cauchy-Schwarz inequality
- Proof of Triangle inequality for vectors and distances in n-dimensional spaces
- Notion of $d_\infty$ – metric

### Completing this course will help you:

- A step forward towards Topology.
- Handling questions on a geometric approach.
- Understanding core concepts of math objects.
- On "joinning the dots" of many concepts.
- Understanding from the roots of problems.

### Who is the course for?

This course appears to be aimed at individuals interested in gaining a comprehensive understanding of Euclidean space and related concepts. While the specific target audience may vary depending on the educational institution or platform offering the course, it is likely designed for students or professionals in mathematics, physics, engineering, computer science, or any other field where a solid understanding of Euclidean space is valuable.

The course seems to cover various fundamental topics, including distances in Euclidean space, metrics, dots, vectors, projections, inner products, and vector and metric spaces. These concepts are foundational to many areas of mathematics and related disciplines, such as linear algebra, geometry, and mathematical physics. Therefore, the course could be suitable for undergraduate or graduate students, researchers, or professionals seeking to deepen their knowledge and skills in these areas.

It’s worth noting that the level of mathematical rigor and prerequisites for the course may vary. Some courses might require prior knowledge of calculus and linear algebra, while others might provide the necessary background as part of the curriculum. It is recommended to review the course description or syllabus for specific details regarding the intended audience and prerequisites.

## Course Content

### Bibliography

- Artin, M. Algebra, Pearson, Second Edition (2017)
- Strang, G. Introduction to Linear Algebra, Cambridge Press, Fifth Edition (2016)
- Stein, E. Functional Analysis, Princeton University Press, 2011
- Shakarchi, R. Real Analysis, Princeton University Press, 2005
- Pugh, C. Mathematical Analysis, Springer, 2015
- Rudin, W. Mathematical Analysis, McGraw-Hill Publishing Company, 1976