1. course web: MA1006: Algebra - Catalogue of Courses
2. syllabus
3. TEXT BOOK
4. Notes:
4.1. missing part
4.1.1. poly:
4.1.1.1. find the roots
4.1.2. complex number
4.1.2.1. Rational Root Theorem: greast common divider of a0 and aN: p/q
4.1.2.2. Every polynomial P(x) of degree n ≥ 1 with complex coefficients is equal to a product of n polynomials of degree one
4.2. complex number
4.2.1. 9.23
4.2.1.1. geomatric features of complex
4.2.1.1.1. argument of compexnumber: theta
4.2.1.1.2. modules
4.2.1.1.3. cormic curve:
4.2.2. 9.26
4.2.2.1. 圆锥曲线的几何含义
4.2.2.1.1. 各参数的几何意义
4.2.2.2. 虚数坐标用极坐标的方法表示
4.2.2.2.1. 注意Argument属于[0，2pi)
4.2.2.2.2. modulus of complex number
4.2.2.3. 四种case
4.2.2.3.1. 虚数坐标到定点的距离为常数
4.2.2.3.2. |z| = Re -Im
4.2.2.3.3. theta = r
4.2.2.3.4. 环形
4.2.3. 9.27
4.2.3.1. De moivre's Theorem:
4.2.3.1.1. 两个虚数乘积 = 模相乘，角度相加
4.2.3.1.2. 补充：三角函数公式 cos n and sin n
4.2.3.1.3. tut
4.2.3.1.4. 泰勒展开
4.2.3.1.5. 开N次方
4.2.4. 9.30
4.2.4.1. Application of the theorem
4.2.4.1.1. 开N次方
4.2.4.1.2. picture 转N次
4.2.4.1.3. cosn(theta)
4.2.4.1.4. sin(n*theta)
4.2.5. 10.2
4.2.5.1. Euler's formula
4.2.5.1.1. exponential function's properties and possible operations
4.3. linear algebra
4.3.1. 10.7
4.3.1.1. introduction
4.3.1.1.1. def: system of linear equation
4.3.1.1.2. pre-aids:
4.3.1.1.3. in 2D graph:
4.3.1.1.4. 3 examples:
4.3.1.1.5. Gaussian elimination
4.3.1.1.6. also works for complex number
4.3.2. 10.10 Matrix
4.3.2.1. terms and related definations:
4.3.2.1.1. augmented matrix: the matrix related with system of equations
4.3.2.1.2. elementary operations:
4.3.2.1.3. row echelon form
4.3.2.2. determinants
4.3.2.2.1. 引子
4.3.2.2.2. 定义
4.3.2.2.3. 计算方法：去掉第一行
4.3.2.2.4. 性质：
4.3.2.3. 10.21 matrix operations:
4.3.2.3.1. rules:
4.3.2.3.2. +
4.3.2.3.3. -
4.3.2.3.4. scalar multiplication
4.3.2.3.5. dot product
4.3.2.3.6. identity matrix has to be square!
4.3.2.3.7. zero matrix
4.3.2.3.8. inverse of matrix
4.4. geometry of the matrices
4.4.1. transportation: Rn -> Rm
4.4.1.1. Lead:
4.4.1.1.1. function can generates tuples
4.4.1.2. transportation
4.4.1.2.1. Rn ->Rm
4.4.1.3. further: linear transformation(f)
4.4.1.3.1. def
4.4.1.3.2. this transportation can be connected with a matrix A
4.4.1.3.3. Proposition
4.4.1.4. standard basis vectors
4.4.1.4.1. transformation of the standard basis vectors makes up the original matrix:
4.4.1.5. composition:
4.4.1.5.1. gf(x) = BA x
4.4.2. certain kinds of transportation:
4.4.2.1. general methods: by studying the transformation of the standard basis vectors
4.4.2.2. determinants and areas
4.4.2.2.1. say there is a unit square
4.4.3. inverse of transformation and inverse of matrix
4.4.3.1. notice that when determinant with 0 is not invertiable, since it collapsed, and can not be saved back
4.5. Eigen Values and Eigen Vecotrs
4.5.1. Def:
4.5.1.1. 10.1. Let A be an (n×n)-matrix. A nonzero vector x ∈ R n is called an eigenvector of A if there exists a number λ ∈ R such that Ax = λx.
4.5.1.1.1. INTERPRETION
4.5.1.1.2. λ is called eigen value(Lam)
4.5.1.1.3. x is called eigen vectors
4.5.2. eigen values and eigen vectors are the concepts based on matrix
4.5.2.1. to study the matrix:
4.5.2.1.1. for scale
4.5.2.1.2. for reflection
4.5.2.1.3. for rotation
4.5.3. How to find eigen values
4.5.3.1. Ax-Lamx = 0
4.5.3.1.1. (A − λI)x = 0.
4.5.3.2. characteristic polynomial
4.5.3.2.1. det(A-Iλ)
4.5.4. how to find eigen vectors
4.5.4.1. if A is Known, Lambda is known, then find x is just solving the linear equation system (A - Lambda*I)x = 0
4.5.4.1.1. possible outcomes
4.5.5. Characteristic of Characteristic Poly:
4.5.5.1. square matrix is invertible iff doesnt have 0 as eigen vectors
4.5.5.1.1. 证：带入Lambda = 0
4.5.5.2. terms of the Polys can be expressed by quantities of Matrix
4.5.5.2.1. general formulas
4.5.5.2.2. 2 by 2
4.5.5.3. Powers of the matrix can be defined with this
4.5.5.3.1. notice that the parameter is pA() can be A itself !
4.6. diagonalization
4.6.1. Lead:
4.6.1.1. in Linear transformation, reflection, notice that T's matrix
4.6.1.1.1. Its actually a change of basis
4.6.2. similarity:
4.6.2.1. B is similar to A if there is an invertible matrix P, where B = P^-1 *A* P
4.6.2.1.1. Proposition 11.2
4.6.3. special cases happens when matrix is "similared" to a diagonal matrix, called diagonalization
4.6.3.1. def: A square matrix A is called diagonalizable if it is similar to a diagonal matrix D. If P is an invertible matrix such that P −1AP = D then P is said to diagonalize A.
4.6.4. criterion for diagonalizability
4.6.4.1. Let A be an (n × n)-matrix. Then A is diagonalizable IFF there is an invertible matrix P = x1 x2 · · · xn， where xi is its eigen vectors
4.6.4.1.1. questions related to
4.6.4.1.2. proof from 2 sides:
4.6.4.2. notice that P is invertible, made up of eigen vectors, the result is eigen values in order
4.6.4.2.1. change the way u constract P
4.6.5. diagonalize the matrix:
4.6.5.1. steps with an example
4.6.5.2. cases when cant diagonalize the matrix: upper matrix, since there is only 1 set of eigen vectors, can not form a matrix with determinant
4.6.6. Linear independant
4.6.6.1. a set of vectors is said to be linearly independent iff : k1 *x + k2*x + ....kn*x = 0, all the k are 0
4.6.6.2. a matrix is only diable iff it has n linearly independent eigen \ values
4.6.7. application:
4.6.7.1. Power of the matrix
4.6.7.1.1. Power of the Diagonal matrix: D^k
4.6.7.1.2. Power of the general: P* D(A)^k*P^-1
4.6.7.2. exponential of the matrix
4.6.7.2.1. e^A = P*e^D*P^-1
4.6.7.3. solution for the first order differential equation
4.6.7.3.1. for the x` = kx
4.6.7.3.2. for the matrix: x` = Ax
5. tut material
5.1. ex 2简单的复数运算
5.1.1. 9, 10
5.2. ex 3 De more, 复杂的复数运算
5.2.1. 10
5.2.1.1. b
5.2.1.1.1. 题目意思为， 所有的项数为实数的poly都可以写成linear与quadratic之积
5.2.1.2. c
5.2.1.2.1. easy
5.2.2. when r = 0, theta is not defined!