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MA1006 by Mind Map: MA1006

1. course web: MA1006: Algebra - Catalogue of Courses

2. syllabus


4. Notes:

4.1. missing part

4.1.1. poly: find the roots

4.1.2. complex number Rational Root Theorem: greast common divider of a0 and aN: p/q 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 geomatric features of complex argument of compexnumber: theta modules cormic curve:

4.2.2. 9.26 圆锥曲线的几何含义 各参数的几何意义 虚数坐标用极坐标的方法表示 注意Argument属于[0,2pi) modulus of complex number 四种case 虚数坐标到定点的距离为常数 |z| = Re -Im theta = r 环形

4.2.3. 9.27 De moivre's Theorem: 两个虚数乘积 = 模相乘,角度相加 补充:三角函数公式 cos n and sin n tut 泰勒展开 开N次方

4.2.4. 9.30 Application of the theorem 开N次方 picture 转N次 cosn(theta) sin(n*theta)

4.2.5. 10.2 Euler's formula exponential function's properties and possible operations

4.3. linear algebra

4.3.1. 10.7 introduction def: system of linear equation pre-aids: in 2D graph: 3 examples: Gaussian elimination also works for complex number

4.3.2. 10.10 Matrix terms and related definations: augmented matrix: the matrix related with system of equations elementary operations: row echelon form determinants 引子 定义 计算方法:去掉第一行 性质: 10.21 matrix operations: rules: + - scalar multiplication dot product identity matrix has to be square! zero matrix inverse of matrix

4.4. geometry of the matrices

4.4.1. transportation: Rn -> Rm Lead: function can generates tuples transportation Rn ->Rm further: linear transformation(f) def this transportation can be connected with a matrix A Proposition standard basis vectors transformation of the standard basis vectors makes up the original matrix: composition: gf(x) = BA x

4.4.2. certain kinds of transportation: general methods: by studying the transformation of the standard basis vectors determinants and areas say there is a unit square

4.4.3. inverse of transformation and inverse of matrix 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: 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. INTERPRETION λ is called eigen value(Lam) x is called eigen vectors

4.5.2. eigen values and eigen vectors are the concepts based on matrix to study the matrix: for scale for reflection for rotation

4.5.3. How to find eigen values Ax-Lamx = 0 (A − λI)x = 0. characteristic polynomial det(A-Iλ)

4.5.4. how to find eigen vectors if A is Known, Lambda is known, then find x is just solving the linear equation system (A - Lambda*I)x = 0 possible outcomes

4.5.5. Characteristic of Characteristic Poly: square matrix is invertible iff doesnt have 0 as eigen vectors 证:带入Lambda = 0 terms of the Polys can be expressed by quantities of Matrix general formulas 2 by 2 Powers of the matrix can be defined with this notice that the parameter is pA() can be A itself !

4.6. diagonalization

4.6.1. Lead: in Linear transformation, reflection, notice that T's matrix Its actually a change of basis

4.6.2. similarity: B is similar to A if there is an invertible matrix P, where B = P^-1 *A* P Proposition 11.2

4.6.3. special cases happens when matrix is "similared" to a diagonal matrix, called diagonalization 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 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 questions related to proof from 2 sides: notice that P is invertible, made up of eigen vectors, the result is eigen values in order change the way u constract P

4.6.5. diagonalize the matrix: steps with an example 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 a set of vectors is said to be linearly independent iff : k1 *x + k2*x +*x = 0, all the k are 0 a matrix is only diable iff it has n linearly independent eigen \ values

4.6.7. application: Power of the matrix Power of the Diagonal matrix: D^k Power of the general: P* D(A)^k*P^-1 exponential of the matrix e^A = P*e^D*P^-1 solution for the first order differential equation for the x` = kx 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 b 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 c easy

5.2.2. when r = 0, theta is not defined!