--- 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)