---
# System prepended metadata

title: A concise introduction of numbers.
tags: [teaching]

---

---
tags: teaching
---
# A concise introduction of numbers.

## Introduction
<!--Traditional Linear Algebra uses a lot of abstract geometric concept such as Kernel, Image, Direct sums as a basic tool for this course. In this book, we are trying to skip those concepts and find a new way to introduce this subject to elementary school students from perspective of numbers.
-->

### Plan 1
- [Insufficient of real numbers](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/ryM47jH_w)
- [Create Numbers by polynomial](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/SyWFXjrOD)
- [Solving polynomials formally](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/H19DJnSOP)
- New numbers generated by polynomials of degree two
    - [Split numbers](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/ByEVAsBdv)
    - [Ramified numbers](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/HkJB0sHdw)
    - [Unramified numbers](https://hackmd.io/@LVcpSNoxQAim3tB5Xaj-gg/B1t30jrOw)
- [Theory of eigenvectors of matrices](/kqrYZpwmRwSoTmG3qGiCgg)


### Plan 2

- Matrix representation of new numbers
- Congruences and remainder representation of new numbers
- Partial Fraction Decomposition and spectural theory


引入新 numbers 会简化证明吗？ JCF 的证明中，一个是根子空间的分解。本质上就是可以求到充分多的特征向量。另一个是特征向量线性独立的特质。（当时用了很多 injective的特性，kernel 和 image特性等）


### Plan 3: Partial Fraction Decomposition

新计划：



- Laplacian Expansion Slides and classical Adjoint and inverse matrix.

- Partial Fraction Decomposition I(split case and higher degree case).

- Complex Number (+ matrix representation of complex numbers)
- Dual Number and Nilpotent Numbers. Pascal's Triangle. Matrix representation of nilpotent numbers. Young Tableau.
- Matrix representation of Numbers and Polynomial division.
- Partial Fraction Decomposition II.
- Theorem on Projections
- Definition of Eigenvalue and Eigenvectors
- Spectural Decomposition of matrices I -- Diagonalizable Case
- Spectural Decomposition of non-diagonalizable cases

- Young Tableau and Jordan canonical form. 
- Applications in Differential Equations.

### Excercises

[Student Ex](/4La8-GgXQHWoc_JN0fvD2w)