Linear Algebra: What are we going to learn? 李宏毅 Hung-yi Lee.

Slides:

Advertisements

Similar presentations

Section 3.2 Solving Equations Using Multiplication & Division Mr. Beltz & Mr. Sparks.

Advertisements

Solving Quadratic Equations by Factoring Algebra I.

Math 015 Section 6.1 Equations. Many students like to remember the process for solving linear equations as: any algebra expressions on each side variable.

EXAMPLE 4 Solve proportions SOLUTION a x 16 = Multiply. Divide each side by 10. a x 16 = = 10 x5 16 = 10 x80 = x8 Write original proportion.

Copy and complete using a review term from the list below.

3-3 Solving Multiplication Equations. Solve Solution GOAL Find the value of the variable that makes the equation TRUE. The value that makes the equation.

Algebra 1: Solving Equations with variables on BOTH sides.

4 step by step on solving linear equations

Algebra Distributive Property. Vocabulary Equivalent expressions: two expressions that have the same output value for every input value Distributive.

MIMO Multiple Input Multiple Output Communications © Omar Ahmad

Solving Systems of Equations: Elimination Method.

ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: First-Order Second-Order N th -Order Computation of the Output Signal.

ECE 8443 – Pattern Recognition ECE 3163 – Signals and Systems Objectives: First-Order Second-Order N th -Order Computation of the Output Signal Transfer.

MTH 161: Introduction To Statistics

Section 4-1: Introduction to Linear Systems. To understand and solve linear systems.

3-2 Solving Equations by Using Addition and Subtraction Objective: Students will be able to solve equations by using addition and subtraction.

Goal: Solve a system of linear equations in two variables by the linear combination method.

2-3 D IRECT V ARIATION Objective: To write and interpret direct variation equations.

Evaluating Algebraic Expressions 1-7 Solving Equations by Adding or Subtracting Preparation for AF4.0 Students solve simple linear equations and inequalities.

3.1 You should be able to solve one-step equations using algebra. Solve for x.

Unit 3, Lesson 6 Constant of Proportionality and Writing Direct Variation Equations.

1. Describe the end behavior of the graph y = 2x 5 – 3x Sketch a graph of 3 rd degree with a zero at -5 (multiplicity 2) and a zero at 0 (multiplicity.

5-5 Solving Quadratic Equations Objectives:  Solve quadratic equations.

Do Now 1) Factor. 3a2 – 26a + 35.

Solving Quadratic Inequalities Algebraically MM2A4. Students will solve quadratic equations and inequalities in one variable.MM2A4. Students will solve.

EXAMPLE 1 Solve by equating exponents Rewrite 4 and as powers with base Solve 4 = x 1 2 x – 3 (2 ) = (2 ) 2 x – 3x – 1– 1 2 = 2 2 x– x + 3 2x =

Math 4030 – 11b Method of Least Squares. Model: Dependent (response) Variable Independent (control) Variable Random Error Objectives: Find (estimated)

2.1 – Linear and Quadratic Equations Linear Equations.

ALGEBRA READINESS LESSON 3-4 Warm Up Lesson 3-4 Warm Up.

Differential Equations Linear Equations with Variable Coefficients.

Solving Quadratic Equations Using the Zero Product Property March 18, 2014.

6-1 Solving Systems by Graphing 6-2 Solving Systems by Substitution 6-3 Solving Systems by Elimination 6-4 Solving Special Systems 6-5 Applying Systems.

Algebra Math 8 May A brain teaser Think of a number. Add three. Find the square of the result. Subtract nine. Divide by the original number. Subtract.

Solving Equations Using Addition or Subtraction Objective: Students will solve linear equations using addition and subtraction.

Direct Variation Chapter 5 Section 2. Objective  Students will write and graph an equation of a direct variation.

Reasoning with linear equations and inequalities Students will understand that equation is a statement of equality between two expression. Students find.

Solving Quadratic Functions by Factoring. Factoring Review.

Objective: Use factoring to solve quadratic equations. Standard(s) being met: 2.8 Algebra and Functions.

2.1 Linear and Quadratic Functions and Modeling Objective: Students will be able to identify specific types of functions, describe properties of those.

Solving Multistep Linear Equations Using Algebra Tiles

Simultaneous Equations 1

SECTION 2-1 : SOLVING ONE STEP EQUATIONS

Learning Target I can solve and graph multi-step, one-variable inequalities using algebraic properties.

Solving Equations by Factoring and Problem Solving

Solving Linear Systems by Graphing

Solving Linear Systems Algebraically

6-3 Solving Systems Using Elimination

Solving Systems Using Elimination

Solving Quadratic Equations by Factoring March 16, 2015

Solving Linear Equations in one Variable

Solve Linear Equations by Elimination

Linear Algebra Lecture 3.

An algebraic expression that defines a function is a function rule.

Solving Equations Containing Decimals

LINEAR EQUATIONS.

Solving Linear Equations with variables on both sides

What are we going to learn?

Algebra EquationsJeopardy

Systems of Equations Solve by Graphing.

Linear System = System of Linear Equations

Warm Up Do Objective 1b, Page 14 in the End of Course Warm Up Booklet

ALGEBRA TWO Section Writing Equations of Lines

LINEAR EQUATIONS.

Solving Equations Using Multiplication and Division

Solving Quadratic Equations by Factoring March 11, 2016

Solving Linear Equations

Algebra Introduction.

Presentation transcript:

Linear Algebra: What are we going to learn? 李宏毅 Hung-yi Lee

What are we going to learn? A system of linear equations ( 多元一次聯立方程式 ) A linear equation a system of linear equations variables coefficients constant term I believe you know how to solve it.

What are we going to learn? A system of linear equations ( 多元一次聯立方程式 ) m equations n variables In this course, m and n can be large

What are we going to learn? Solving a system of linear equations Does it have solution? Does it have unique solution? How to find the solution? Different view from high school Determinants ( 行列式 ) Beyond 3 X 3 Approximate Solution span Independent rank

Linear System We use a system of linear equations to describe a linear system What is system? function, transformation, operator A system has input and output E.g. Speech Recognition System: The input is audio signals, and the output is recognition results. E.g. Siri: You say “Hi, Siri”, it reply “What can I do for you?” E.g. Communication system Can have multiple inputs and outputs System x1x1 x2x2 y1y1 y2y2

Linear System Linear system have two properties 1. Persevering Multiplication 2. Persevering Addition Linear System xy Linear System kxky x2x2 y2y2 x 1 +x 2 y 1 +y 2 Linear System x1x1 y1y1 Linear System Linear System

Linear System Linear system have two properties 1. Persevering Multiplication 2. Persevering Addition System xx2x2 Linear System x Linear System kx x2x2 k2x2k2x2 Linear System Linear System Linear System

Linear System Two properties 1. Persevering Multiplication 2. Persevering Addition System

Linear System v.s. System of Linear Equations Linear System ? ……

Linear System …… Linear System …… Linear System …… Linear System …… A linear system is described by a system of linear equations

Applications + - input output ( 大二上 ) Most system are (assumed to) be linear Circuit Linear Input: voltage source, current source output: voltage and current on the load ( 燈泡、引擎 ) Linear System

Applications ( 大二下 ) Most system are (assumed to) be linear Linear Transmitter Receiver Communication System Linear System Channel

Applications Machine Learning E.g. Prediction 股票分析坪數氣象預報房價分析位置 Linear system 價格 ……. 今天天氣昨天天氣 ……. 明天天氣今天行情明天行情 Linear system

Applications We will learn EigenXXXXX Filter Linear System …… Input equals to outputOutput is zero Linear System ……

Applications – Image Compression 壓縮比 : 25 壓縮比 : 12.5 壓縮比 : 5 壓縮比 : 2.5 原圖

Linear Algebra is Important