# MA1006

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## 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

4.4.1.1.1. 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

4.6.1.1. 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

5.1.1. 9, 10

5.2.1. 10

5.2.1.1. b