# CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

Size: px
Start display at page:

Download "CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer"

Transcription

1 CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2016

2 Announcements Monday October 3: Discussion Assignment 1 Friday October 7: Assignment 1 due 2

3 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 3

4 Vectors Give direction and length in 3D Vectors can describe Difference between two 3D points Speed of an object Surface normals (directions perpendicular to surfaces) Normal vector Surface normals Surface 4

5 Vector arithmetic using coordinates a = ax ay az b = bx by bz a + b = ax + bx ay + by az + bz a b = ax bx ay by az bz ax a= ay az sax sa= say saz where s is a scalar 5

6 Vector Magnitude The magnitude (length) of a vector is: v 2 = v x 2 + v y 2 + v z 2 v = v x 2 + v y 2 + v z 2 A vector with length of 1.0 is called unit vector We can also normalize a vector to make it a unit vector Unit vectors are often used as surface normals v v 6

7 Dot Product a b = a i b i a b = a x b x + a y b y + a z b z a b = a b cosθ 7

8 Angle Between Two Vectors a b = a b cosθ cosθ = a b a b b θ = cos 1 a b a b a 8

9 Cross Product a b is a vector perpendicular to both a and b, in the direction defined by the right hand rule a b = a b sinθ a b = area of parallelogram ab a b = 0 if aand b are parallel (or one or both degenerate) 9

10 Cross Product = = 10

11 Cross Product Calculation = = = 11

12 Matrices Rectangular array of numbers Square matrix if m = n In graphics almost always: m = n = 3; m = n = 4 12

14 Multiplication With Scalar 14

15 Matrix Multiplication 15

16 Matrix-Vector Multiplication 16

17 Identity Matrix 17

18 Matrix Inverse If a square matrix M is non-singular, there exists a unique inverse M -1 such that 18

19 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 19

20 Affine Transformations Most important for graphics: rotation, translation, scaling Wolfram MathWorld: An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). Implemented using matrix multiplications 20

21 Uniform Scale Uniform scale matrix in 2D Analogous in 3D:

22 Non-Uniform Scale Nonuniform scaling matrix in 2D Analogous in 3D: 22

23 Rotation in 2D Convention: positive angle rotates counterclockwise Rotation matrix 23

24 Rotation in 3D Rotation around coordinate axes 24

25 Rotation in 3D Concatenation of rotations around x, y, z axes are called Euler angles Result depends on matrix order! 25

26 Rotation about an Arbitrary Axis Complicated! Rotate point [x,y,z] about axis [u,v,w] by angle θ: 26

27 How to rotate around a Pivot Point? Rotation around origin: p = R p Rotation around pivot point: p =? 27

28 Rotating point p around a pivot point 1. Translation T 2. Rotation R 3. Translation T -1 p = T -1 R T p 28

29 Concatenating transformations Given a sequence of transformations M 3 M 2 M 1 Note: associativity applies 29

30 Today s Topics Vectors and matrices Affine transformations Homogeneous coordinates 30

31 Translation Translation in 2D t x t y Translation matrix T=? = = = =???? 31

32 Translation Translation in 2D: 3x3 matrix Analogous in 3D: 4x4 matrix 32

33 Homogeneous Coordinates Basic: a trick to unify/simplify computations. Deeper: projective geometry Interesting mathematical properties Good to know, but less immediately practical We will use some aspect of this when we do perspective projection 33

34 Homogeneous Coordinates Add an extra component. 1 for a point, 0 for a vector: p = Combine M and d into single 4x4 matrix: p x p y p z 1 And see what happens when we multiply r v = v x v y v z 0 m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z

35 Homogeneous Point Transform Transform a point: p x p y p z 1 = m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z p x p y p z 1 = m xx p x + m xy p y + m xz p z + d x m yx p x + m yy p y + m yz p z + d y m zx p x + m zy p y + m zz p z + d z Top three rows are the affine transform! Bottom row stays 1 M p x p y p z + r d 35

36 Homogeneous Vector Transform Transform a vector: v x v y v z 0 = m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z v x v y v z 0 = m xx v x + m xy v y + m xz v z + 0 m yx v x + m yy v y + m yz v z + 0 m zx v x + m zy v y + m zz v z M v x v y v z Top three rows are the linear transform Displacement d is properly ignored Bottom row stays 0 36

37 Homogeneous Arithmetic Legal operations always end in 0 or 1! M vector+vector: 0 + M M 0 0 M vector-vector: 0 M M 0 0 scalar*vector: s M M 0 0 M point+vector: 1 + M M 0 1 M point-point: 1 M M 1 0 M point+point: 1 + M M 1 2 scalar*point: s M M 1 s weighted average affine combination of points: 1 M M 3 M

38 Homogeneous Transforms Rotation, Scale, and Translation of points and vectors unified in a single matrix transformation: p = M p Matrix has the form: Last row always 0,0,0,1 m xx m xy m xz d x m yx m yy m yz d y m zx m zy m zz d z Transforms can be composed by matrix multiplication Same caveat: order of operations is important Same note: transforms operate right-to-left 38

### CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer

CSE 167: Introduction to Computer Graphics Lecture #2: Linear Algebra Primer Jürgen P. Schulze, Ph.D. University of California, San Diego Spring Quarter 2016 Announcements Project 1 due next Friday at

### COMP 175 COMPUTER GRAPHICS. Lecture 04: Transform 1. COMP 175: Computer Graphics February 9, Erik Anderson 04 Transform 1

Lecture 04: Transform COMP 75: Computer Graphics February 9, 206 /59 Admin Sign up via email/piazza for your in-person grading Anderson@cs.tufts.edu 2/59 Geometric Transform Apply transforms to a hierarchy

### 11.1 Three-Dimensional Coordinate System

11.1 Three-Dimensional Coordinate System In three dimensions, a point has three coordinates: (x,y,z). The normal orientation of the x, y, and z-axes is shown below. The three axes divide the region into

### Mathematics for 3D Graphics

math 1 Topics Mathematics for 3D Graphics math 1 Points, Vectors, Vertices, Coordinates Dot Products, Cross Products Lines, Planes, Intercepts References Many texts cover the linear algebra used for 3D

### A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#\$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y \$ Z % Y Y x x } / % «] «] # z» & Y X»

### GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS

GG303 Lecture 6 8/27/09 1 SCALARS, VECTORS, AND TENSORS I Main Topics A Why deal with tensors? B Order of scalars, vectors, and tensors C Linear transformation of scalars and vectors (and tensors) II Why

### 1 Overview. CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992

CS348a: Computer Graphics Handout #8 Geometric Modeling Original Handout #8 Stanford University Thursday, 15 October 1992 Original Lecture #1: 1 October 1992 Topics: Affine vs. Projective Geometries Scribe:

### In this section, mathematical description of the motion of fluid elements moving in a flow field is

Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

### Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems. Three-Dimensional Coordinate Systems

To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate.

### LECTURE 5, FRIDAY

LECTURE 5, FRIDAY 20.02.04 FRANZ LEMMERMEYER Before we start with the arithmetic of elliptic curves, let us talk a little bit about multiplicities, tangents, and singular points. 1. Tangents How do we

### 1.1 Single Variable Calculus versus Multivariable Calculus Rectangular Coordinate Systems... 4

MATH2202 Notebook 1 Fall 2015/2016 prepared by Professor Jenny Baglivo Contents 1 MATH2202 Notebook 1 3 1.1 Single Variable Calculus versus Multivariable Calculus................... 3 1.2 Rectangular Coordinate

### CS 4300 Computer Graphics. Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011

CS 4300 Computer Graphics Prof. Harriet Fell Fall 2011 Lecture 11 September 29, 2011 October 8, 2011 College of Computer and Information Science, Northeastern Universit 1 Toda s Topics Linear Algebra Review

### If the pull is downward (Fig. 1), we want C to point into the page. If the pull is upward (Fig. 2), we want C to point out of the page.

11.5 Cross Product Contemporary Calculus 1 11.5 CROSS PRODUCT This section is the final one about the arithmetic of vectors, and it introduces a second type of vector vector multiplication called the cross

### Homogeneous Transformations

Purpose: Homogeneous Transformations The purpose of this chapter is to introduce you to the Homogeneous Transformation. This simple 4 x 4 transformation is used in the geometry engines of CAD systems and

### Matrix-Vector Products and the Matrix Equation Ax = b

Matrix-Vector Products and the Matrix Equation Ax = b A. Havens Department of Mathematics University of Massachusetts, Amherst January 31, 2018 Outline 1 Matrices Acting on Vectors Linear Combinations

### Properties of surfaces II: Second moment of area

Properties of surfaces II: Second moment of area Just as we have discussing first moment of an area and its relation with problems in mechanics, we will now describe second moment and product of area of

### Lecture Notes for MATH6106. March 25, 2010

Lecture Notes for MATH66 March 25, 2 Contents Vectors 4. Points in Space.......................... 4.2 Distance between Points..................... 4.3 Scalars and Vectors........................ 5.4 Vectors

### Omm Al-Qura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics

LECTURE OUTLINE CHAPTER 3 Vectors in Physics 3-1 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector

### Exercise 1: Inertia moment of a simple pendulum

Exercise : Inertia moment of a simple pendulum A simple pendulum is represented in Figure. Three reference frames are introduced: R is the fixed/inertial RF, with origin in the rotation center and i along

### Lecture 8: Coordinate Frames. CITS3003 Graphics & Animation

Lecture 8: Coordinate Frames CITS3003 Graphics & Animation E. Angel and D. Shreiner: Interactive Computer Graphics 6E Addison-Wesley 2012 Objectives Learn how to define and change coordinate frames Introduce

### Classical Mechanics. Luis Anchordoqui

1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body

### Announcements Wednesday, September 27

Announcements Wednesday, September 27 The midterm will be returned in recitation on Friday. You can pick it up from me in office hours before then. Keep tabs on your grades on Canvas. WeBWorK 1.7 is due

### MAE 323: Lecture 1. Review

This review is divided into two parts. The first part is a mini-review of statics and solid mechanics. The second part is a review of matrix/vector fundamentals. The first part is given as an refresher

### Problem Set 2 Due Tuesday, September 27, ; p : 0. (b) Construct a representation using five d orbitals that sit on the origin as a basis: 1

Problem Set 2 Due Tuesday, September 27, 211 Problems from Carter: Chapter 2: 2a-d,g,h,j 2.6, 2.9; Chapter 3: 1a-d,f,g 3.3, 3.6, 3.7 Additional problems: (1) Consider the D 4 point group and use a coordinate

### Vectors for Physics. AP Physics C

Vectors for Physics AP Physics C A Vector is a quantity that has a magnitude (size) AND a direction. can be in one-dimension, two-dimensions, or even three-dimensions can be represented using a magnitude

### MAC Module 5 Vectors in 2-Space and 3-Space II

MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 1 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in R 3. 2. Determine a scalar triple

### Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

### LOWELL WEEKLY JOURNAL

Y -» \$ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - \$ { Q» / X x»»- 3 q \$ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

### MSMS Vectors and Matrices

MSMS Vectors and Matrices Basilio Bona DAUIN Politecnico di Torino Semester 1, 2015-2016 B. Bona (DAUIN) MSMS-Vectors and matrices Semester 1, 2015-2016 1 / 39 Introduction Most of the topics introduced

### 6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant

DE2-EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents Three-Dimensional

MTH 310-3 Abstract Algebra I and Number Theory S18 Homework 1/Solutions Graded Exercises Exercise 1. Below are parts of the addition table and parts of the multiplication table of a ring. Complete both

### CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra /34

Linear Algebra /34 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct axes)

### MAT 419 Lecture Notes Transcribed by Eowyn Cenek 6/1/2012

(Homework 1: Chapter 1: Exercises 1-7, 9, 11, 19, due Monday June 11th See also the course website for lectures, assignments, etc) Note: today s lecture is primarily about definitions Lots of definitions

### Problem Set 2 Due Thursday, October 1, & & & & # % (b) Construct a representation using five d orbitals that sit on the origin as a basis:

Problem Set 2 Due Thursday, October 1, 29 Problems from Cotton: Chapter 4: 4.6, 4.7; Chapter 6: 6.2, 6.4, 6.5 Additional problems: (1) Consider the D 3h point group and use a coordinate system wherein

### 03 - Basic Linear Algebra and 2D Transformations

03 - Basic Linear Algebra and 2D Transformations (invited lecture by Dr. Marcel Campen) Overview In this box, you will find references to Eigen We will briefly overview the basic linear algebra concepts

### Chapter 2. Vectors and Vector Spaces

2.2. Cartesian Coordinates and Geometrical Properties of Vectors 1 Chapter 2. Vectors and Vector Spaces Section 2.2. Cartesian Coordinates and Geometrical Properties of Vectors Note. There is a natural

### Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups

Bidiagonal pairs, Tridiagonal pairs, Lie algebras, and Quantum Groups Darren Funk-Neubauer Department of Mathematics and Physics Colorado State University - Pueblo Pueblo, Colorado, USA darren.funkneubauer@colostate-pueblo.edu

### Announcements Monday, September 18

Announcements Monday, September 18 WeBWorK 1.4, 1.5 are due on Wednesday at 11:59pm. The first midterm is on this Friday, September 22. Midterms happen during recitation. The exam covers through 1.5. About

### 1. Vectors.

1. Vectors 1.1 Vectors and Matrices Linear algebra is concerned with two basic kinds of quantities: vectors and matrices. 1.1 Vectors and Matrices Scalars and Vectors - Scalar: a numerical value denoted

### ALGEBRAIC GEOMETRY HOMEWORK 3

ALGEBRAIC GEOMETRY HOMEWORK 3 (1) Consider the curve Y 2 = X 2 (X + 1). (a) Sketch the curve. (b) Determine the singular point P on C. (c) For all lines through P, determine the intersection multiplicity

### Detailed objectives are given in each of the sections listed below. 1. Cartesian Space Coordinates. 2. Displacements, Forces, Velocities and Vectors

Unit 1 Vectors In this unit, we introduce vectors, vector operations, and equations of lines and planes. Note: Unit 1 is based on Chapter 12 of the textbook, Salas and Hille s Calculus: Several Variables,

### Algebraic Expressions

Algebraic Expressions 1. Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly

### 12. Stresses and Strains

12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

### CHAPTER 4 Stress Transformation

CHAPTER 4 Stress Transformation ANALYSIS OF STRESS For this topic, the stresses to be considered are not on the perpendicular and parallel planes only but also on other inclined planes. A P a a b b P z

### Transformations. Chapter D Transformations Translation

Chapter 4 Transformations Transformations between arbitrary vector spaces, especially linear transformations, are usually studied in a linear algebra class. Here, we focus our attention to transformation

### Chapter 2 Math Fundamentals

Chapter 2 Math Fundamentals Part 1 2.1 Conventions and Definitions 2.2 Matrices 2.3 Fundamentals of Rigid Transforms 1 Outline 2.1 Conventions and Definitions 2.2 Matrices 2.3 Fundamentals of Rigid Transforms

### Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

### Matrix Arithmetic. a 11 a. A + B = + a m1 a mn. + b. a 11 + b 11 a 1n + b 1n = a m1. b m1 b mn. and scalar multiplication for matrices via.

Matrix Arithmetic There is an arithmetic for matrices that can be viewed as extending the arithmetic we have developed for vectors to the more general setting of rectangular arrays: if A and B are m n

### A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010

A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics

### LECTURE 10, MONDAY MARCH 15, 2004

LECTURE 10, MONDAY MARCH 15, 2004 FRANZ LEMMERMEYER 1. Minimal Polynomials Let α and β be algebraic numbers, and let f and g denote their minimal polynomials. Consider the resultant R(X) of the polynomials

### x 9 or x > 10 Name: Class: Date: 1 How many natural numbers are between 1.5 and 4.5 on the number line?

1 How many natural numbers are between 1.5 and 4.5 on the number line? 2 How many composite numbers are between 7 and 13 on the number line? 3 How many prime numbers are between 7 and 20 on the number

### Contents. 1 Vectors, Lines and Planes 1. 2 Gaussian Elimination Matrices Vector Spaces and Subspaces 124

Matrices Math 220 Copyright 2016 Pinaki Das This document is freely redistributable under the terms of the GNU Free Documentation License For more information, visit http://wwwgnuorg/copyleft/fdlhtml Contents

### Problem 1: (3 points) Recall that the dot product of two vectors in R 3 is

Linear Algebra, Spring 206 Homework 3 Name: Problem : (3 points) Recall that the dot product of two vectors in R 3 is a x b y = ax + by + cz, c z and this is essentially the same as the matrix multiplication

### 6. SCALARS, VECTORS, AND TENSORS (FOR ORTHOGONAL COORDINATE SYSTEMS)

(FOR ORTHOGONAL COORDINATE SYSTEMS) I Main Topics A What are scalars, vectors, and tensors? B Order of scalars, vectors, and tensors C Linear transformaoon of scalars and vectors (and tensors) D Matrix

### ARC 341 Structural Analysis II. Lecture 10: MM1.3 MM1.13

ARC241 Structural Analysis I Lecture 10: MM1.3 MM1.13 MM1.4) Analysis and Design MM1.5) Axial Loading; Normal Stress MM1.6) Shearing Stress MM1.7) Bearing Stress in Connections MM1.9) Method of Problem

### Neatest and Promptest Manner. E d i t u r ami rul)lihher. FOIt THE CIIILDIIES'. Trifles.

» ~ \$ ) 7 x X ) / ( 8 2 X 39 ««x» ««! «! / x? \» «({? «» q «(? (?? x! «? 8? ( z x x q? ) «q q q ) x z x 69 7( X X ( 3»«! ( ~«x ««x ) (» «8 4 X «4 «4 «8 X «x «(» X) ()»» «X «97 X X X 4 ( 86) x) ( ) z z

### Review of Coordinate Systems

Vector in 2 R and 3 R Review of Coordinate Systems Used to describe the position of a point in space Common coordinate systems are: Cartesian Polar Cartesian Coordinate System Also called rectangular coordinate

### Department of Physics, Korea University

Name: Department: Notice +2 ( 1) points per correct (incorrect) answer. No penalty for an unanswered question. Fill the blank ( ) with (8) if the statement is correct (incorrect).!!!: corrections to an

### Linear Algebra & Geometry why is linear algebra useful in computer vision?

Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia

### CSC 470 Introduction to Computer Graphics. Mathematical Foundations Part 2

CSC 47 Introduction to Computer Graphics Mathematical Foundations Part 2 Vector Magnitude and Unit Vectors The magnitude (length, size) of n-vector w is written w 2 2 2 w = w + w2 + + w n Example: the

### Rigid Body Rotation. Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li. Department of Applied Mathematics and Statistics Stony Brook University (SUNY)

Rigid Body Rotation Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied Mathematics and Statistics Stony Brook University (SUNY) Content Introduction Angular Velocity Angular Momentum

### Chapter 1. Vector Analysis

Chapter 1. Vector Analysis Hayt; 8/31/2009; 1-1 1.1 Scalars and Vectors Scalar : Vector: A quantity represented by a single real number Distance, time, temperature, voltage, etc Magnitude and direction

### Homogeneous Coordinates

Homogeneous Coordinates Basilio Bona DAUIN-Politecnico di Torino October 2013 Basilio Bona (DAUIN-Politecnico di Torino) Homogeneous Coordinates October 2013 1 / 32 Introduction Homogeneous coordinates

### Lesson Rigid Body Dynamics

Lesson 8 Rigid Body Dynamics Lesson 8 Outline Problem definition and motivations Dynamics of rigid bodies The equation of unconstrained motion (ODE) User and time control Demos / tools / libs Rigid Body

### Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

### The geometry of least squares

The geometry of least squares We can think of a vector as a point in space, where the elements of the vector are the coordinates of the point. Consider for example, the following vector s: t = ( 4, 0),

### Sec. 1.1: Basics of Vectors

Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x

### MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

### (A B) 2 + (A B) 2. and factor the result.

Transformational Geometry of the Plane (Master Plan) Day 1. Some Coordinate Geometry. Cartesian (rectangular) coordinates on the plane. What is a line segment? What is a (right) triangle? State and prove

### Rotational & Rigid-Body Mechanics. Lectures 3+4

Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions

### Last week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v

Orthogonality (I) Last week we presented the following expression for the angles between two vectors u and v in R n ( ) θ = cos 1 u v u v which brings us to the fact that θ = π/2 u v = 0. Definition (Orthogonality).

### Linear Algebra V = T = ( 4 3 ).

Linear Algebra Vectors A column vector is a list of numbers stored vertically The dimension of a column vector is the number of values in the vector W is a -dimensional column vector and V is a 5-dimensional

### CS123 INTRODUCTION TO COMPUTER GRAPHICS. Linear Algebra 1/33

Linear Algebra 1/33 Vectors A vector is a magnitude and a direction Magnitude = v Direction Also known as norm, length Represented by unit vectors (vectors with a length of 1 that point along distinct

### Lesson 6. Diana Pell. Monday, March 17. Section 4.1: Solve Linear Inequalities Using Properties of Inequality

Lesson 6 Diana Pell Monday, March 17 Section 4.1: Solve Linear Inequalities Using Properties of Inequality Example 1. Solve each inequality. Graph the solution set and write it using interval notation.

### I&C 6N. Computational Linear Algebra

I&C 6N Computational Linear Algebra 1 Lecture 1: Scalars and Vectors What is a scalar? Computer representation of a scalar Scalar Equality Scalar Operations Addition and Multiplication What is a vector?

### Math 51, Homework-2 Solutions

SSEA Summer 27 Math 5, Homework-2 Solutions Write the parametric equation of the plane that contains the following point and line: 3 2, 4 2 + t 3 t R 5 4 By substituting t = and t =, we get two points

### The Sphere OPTIONAL - I Vectors and three dimensional Geometry THE SPHERE

36 THE SPHERE You must have played or seen students playing football, basketball or table tennis. Football, basketball, table tennis ball are all examples of geometrical figures which we call "spheres"

### Algebraic Expressions and Identities

ALGEBRAIC EXPRESSIONS AND IDENTITIES 137 Algebraic Expressions and Identities CHAPTER 9 9.1 What are Expressions? In earlier classes, we have already become familiar with what algebraic expressions (or

### LOWELL JOURNAL. MUST APOLOGIZE. such communication with the shore as Is m i Boimhle, noewwary and proper for the comfort

- 7 7 Z 8 q ) V x - X > q - < Y Y X V - z - - - - V - V - q \ - q q < -- V - - - x - - V q > x - x q - x q - x - - - 7 -» - - - - 6 q x - > - - x - - - x- - - q q - V - x - - ( Y q Y7 - >»> - x Y - ] [

### Math Exam 2, October 14, 2008

Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian

### Vector and Affine Math

Vector and Affine Math Computer Science Department The Universit of Teas at Austin Vectors A vector is a direction and a magnitude Does NOT include a point of reference Usuall thought of as an arrow in

### A geometric interpretation of the homogeneous coordinates is given in the following Figure.

Introduction Homogeneous coordinates are an augmented representation of points and lines in R n spaces, embedding them in R n+1, hence using n + 1 parameters. This representation is useful in dealing with

### Linear Algebra Homework and Study Guide

Linear Algebra Homework and Study Guide Phil R. Smith, Ph.D. February 28, 20 Homework Problem Sets Organized by Learning Outcomes Test I: Systems of Linear Equations; Matrices Lesson. Give examples of

### (arrows denote positive direction)

12 Chapter 12 12.1 3-dimensional Coordinate System The 3-dimensional coordinate system we use are coordinates on R 3. The coordinate is presented as a triple of numbers: (a,b,c). In the Cartesian coordinate

### Hilbert s Metric and Gromov Hyperbolicity

Hilbert s Metric and Gromov Hyperbolicity Andrew Altman May 13, 2014 1 1 HILBERT METRIC 2 1 Hilbert Metric The Hilbert metric is a distance function defined on a convex bounded subset of the n-dimensional

### 7a3 2. (c) πa 3 (d) πa 3 (e) πa3

1.(6pts) Find the integral x, y, z d S where H is the part of the upper hemisphere of H x 2 + y 2 + z 2 = a 2 above the plane z = a and the normal points up. ( 2 π ) Useful Facts: cos = 1 and ds = ±a sin

### Two Posts to Fill On School Board

Y Y 9 86 4 4 qz 86 x : ( ) z 7 854 Y x 4 z z x x 4 87 88 Y 5 x q x 8 Y 8 x x : 6 ; : 5 x ; 4 ( z ; ( ) ) x ; z 94 ; x 3 3 3 5 94 ; ; ; ; 3 x : 5 89 q ; ; x ; x ; ; x : ; ; ; ; ; ; 87 47% : () : / : 83

### A = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].

Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes

### Unit 2: Lines and Planes in 3 Space. Linear Combinations of Vectors

Lesson10.notebook November 28, 2012 Unit 2: Lines and Planes in 3 Space Linear Combinations of Vectors Today's goal: I can write vectors as linear combinations of each other using the appropriate method

### Vectors and Matrices

Vectors and Matrices Scalars We often employ a single number to represent quantities that we use in our daily lives such as weight, height etc. The magnitude of this number depends on our age and whether

### Lagrange Multipliers

Optimization with Constraints As long as algebra and geometry have been separated, their progress have been slow and their uses limited; but when these two sciences have been united, they have lent each

### Lecture 4: Affine Transformations. for Satan himself is transformed into an angel of light. 2 Corinthians 11:14

Lecture 4: Affine Transformations for Satan himself is transformed into an angel of light. 2 Corinthians 11:14 1. Transformations Transformations are the lifeblood of geometry. Euclidean geometry is based

### Intro Vectors 2D implicit curves 2D parametric curves. Graphics 2012/2013, 4th quarter. Lecture 2: vectors, curves, and surfaces

Lecture 2, curves, and surfaces Organizational remarks Tutorials: TA sessions for tutorial 1 start today Tutorial 2 will go online after lecture 3 Practicals: Make sure to find a team partner very soon

### MCE/EEC 647/747: Robot Dynamics and Control. Lecture 2: Rigid Motions and Homogeneous Transformations

MCE/EEC 647/747: Robot Dynamics and Control Lecture 2: Rigid Motions and Homogeneous Transformations Reading: SHV Chapter 2 Mechanical Engineering Hanz Richter, PhD MCE503 p.1/22 Representing Points, Vectors

### Math 302 Test 1 Review

Math Test Review. Given two points in R, x, y, z and x, y, z, show the point x + x, y + y, z + z is on the line between these two points and is the same distance from each of them. The line is rt x, y,

### Course 2BA1: Hilary Term 2007 Section 8: Quaternions and Rotations

Course BA1: Hilary Term 007 Section 8: Quaternions and Rotations David R. Wilkins Copyright c David R. Wilkins 005 Contents 8 Quaternions and Rotations 1 8.1 Quaternions............................ 1 8.

### Chapter 2: Numeric, Cell, and Structure Arrays

Chapter 2: Numeric, Cell, and Structure Arrays Topics Covered: Vectors Definition Addition Multiplication Scalar, Dot, Cross Matrices Row, Column, Square Transpose Addition Multiplication Scalar-Matrix,

### Lecture 3: Linear Algebra Review, Part II

Lecture 3: Linear Algebra Review, Part II Brian Borchers January 4, Linear Independence Definition The vectors v, v,..., v n are linearly independent if the system of equations c v + c v +...+ c n v n