also today: how to read a research paper CS 4300/5310 Computer Graphics

Size: px
Start display at page:

Download "also today: how to read a research paper CS 4300/5310 Computer Graphics"

Transcription

1 also toda: how to read a research paper CS 4300/5310 Computer Graphics

2 ANNOUNCEMENTS

3 Deadlines 2D Project Proposal: Januar 22 nd Submit one per group 2D Project main deadline: Februar 5 th Reading Response: Januar 22 nd It s short Don t worr

4 Global Game Jam

5 Game Demo Da Submission deadline: March 29 Event: April 19 Campus wide demo event for games developed during Industr judges Great to add to resume

6 MATRIX MATH: QUICK REVIEW

7 Matrices A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 m n (3 3) (rows columns)

8 Matri Square matri m=n Diagonal Matri a ij = 0 if i j A = a a a 33 Zero Matri all a ij =0

9 Matri Square matri m=n Diagonal Matri a ij = 0 if i j A = a a a 33 Zero Matri all a ij =0

10 Matri Square matri m=n Diagonal Matri a ij = 0 if i j A = a a a 33 Zero Matri all a ij =0

11 Matri Square matri m=n Diagonal Matri a ij = 0 if i j A = a a a 33 Zero Matri all a ij =0

12 Matri addigon A + B = C a 11 a 12 a 21 a 22 + b 11 b 12 b 21 b 22 = a 11 + b 11 a 12 + b 12 a 21 + b 21 a 22 + b = Constraint: m A =m B and n A =n B

13 Matri SubtracGon A B = C a 11 a 12 a 21 a 22 b 11 b 12 b 21 b 22 = a 11 b 11 a 12 b 12 a 21 b 21 a 22 b = Constraint: m A =m B and n A =n B

14 Matri Scalar MulGplicaGon ba = C a 11 a 12 b a 21 a 22 = b a 11 b a 12 b a 21 b a =

15 Matri MulGplicaGon n AB = C, c ij = a ik b kj (m A n B ) a 11 a 12 a 21 a 22 a 31 a 32 ( ( ( k=1 b 11 b 12 b 21 b 22 ( = a 11 b 11 + a 12 b 21 a 11 b 12 + a 12 b 22 a 21 b 11 + a 22 b 21 a 21 b 12 + a 22 b 22 a 31 b 11 + a 32 b 21 a 31 b 12 + a 32 b 22 ( ((32matri) ( = Constraint: n A = m B

16 Matri MulGplicaGon DistribuUve: A(B+C) = AB + AC AssociaUve: (AB)C = A(BC) Not commutauve: AB is not equal to BA

17 Matri Transpose A T, a ij becomes a ji A = A T = (3 2) (2 3)

18 2D TRANSFORMATIONS

19 Transforms a 11 a 12 a 12 a 22 TransformaGon Matri

20 TransformaGon: Scaling s 0???? 0 s

21 TransformaGon: Scaling s 0 0 s

22 TransformaGon: ProjecGon = = 0

23 TransformaGon: ReflecGon =

24 TransformaGon: ReflecGon ReflecUon over = line ???? =

25 TransformaGon: ReflecGon ReflecUon over = line =

26 TransformaGon: Shearing ais = + ais = +

27 TransformaGon: RotaGon cosθ sinθ clockwise sinθ cosθ cosθ counter clockwise sinθ sinθ cosθ

28 TransformaGon: ComposiGon Order is ver important Read right le_. What does this do? s 0 0 s

29 TranslaGon RotaUon, scaling, shearing, etc. are linear transformauons a 11 a 12 a 21 a 22 = a 11 + a 12 a 21 + a 22 But we want: + t + t

30 Changing our representagon Represent the point, b vector Thus: 1 1 = 1 0 t 0 1 t ( ) +, 1 ( ) +, = + t + t 1 ( ) +, Homogenous Coordinates

31 Homogeneous Coordinates Vectors vs. Points: Thus: = a 11 a 12 t a 21 a 22 t ( ) +, 0 ( ) +, = a 11 + a 12 a 21 + a 22 0 ( ) +, Points Vectors

32 TransformaGon Matri Use same matri for scaling, rotauon, etc. Eample: 1 = a 11 0 t 0 a 22 t ( ) +, 1 ( ) +, = a 11 + t a 22 + t 1 ( ) +,

33 Scene Graphs

34 HOW TO READ A RESEARCH PAPER

35 Step One: Authors are Human Authors are people like ou Read crigcall don t assume the are correct Research papers are peer reviewed

36 Step Two: Structure of a Paper IntroducUon Related Work Method/Approach EvaluaUon Conclusions References

37 Step Three: Reading CriGcall IntroducUon What problem are the solving? Does it make sense to solve it? Is there a beger problem? Is the problem oversimplified? Do ou agree with their arguments?

38 Step Three: Reading CriGcall Related Work Is the related work actuall related? Are the comparing appropriatel? Are the describing the other work fairl?

39 Step Three: Reading CriGcall Method/Approach Is there enough detail? Does this approach make sense? Wh are the making these decisions? What assumpuons are being made? Could this be improved? But what if I reall don t understand? Flag concepts ou don t understand Go to the references Ask quesuons on piazza/in class

40 Step Three: Reading CriGcall EvaluaUon Are ou convinced b the results? Are the tesung their approach appropriatel? What further informauon do ou wish ou had?

41 Step Three: Reading CriGcall Conclusion Are the authors drawing the right conclusion? Does the future work make sense? Do the authors make their claims about what the ve done clear?

42 Step Four: Reading Creavel What would I do differentl? How would I etend the work presented? How would I evaluate it differentl? What do I think the impact of this could be on other areas I m interested in? If I were to start working on a project in this area, what s the first thing I would do net?

43 Reading Responses 1 page Brief summar of the paper Aim for no more than 2 3 sentences What problem were the solving? What did the learn? The rest is our opinion What did/didn t ou like in the paper? What quesuons did ou ask ourself when reading?

CS 378: Computer Game Technology

CS 378: Computer Game Technology CS 378: Computer Game Technolog 3D Engines and Scene Graphs Spring 202 Universit of Teas at Austin CS 378 Game Technolog Don Fussell Representation! We can represent a point, p =,), in the plane! as a

More information

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations

Kevin James. MTHSC 3110 Section 2.1 Matrix Operations MTHSC 3110 Section 2.1 Matrix Operations Notation Let A be an m n matrix, that is, m rows and n columns. We ll refer to the entries of A by their row and column indices. The entry in the i th row and j

More information

Matrices and Vectors

Matrices and Vectors Matrices and Vectors James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University November 11, 2013 Outline 1 Matrices and Vectors 2 Vector Details 3 Matrix

More information

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages:

CS100: DISCRETE STRUCTURES. Lecture 3 Matrices Ch 3 Pages: CS100: DISCRETE STRUCTURES Lecture 3 Matrices Ch 3 Pages: 246-262 Matrices 2 Introduction DEFINITION 1: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n

More information

Matrix Multiplication

Matrix Multiplication 3.2 Matrix Algebra Matrix Multiplication Example Foxboro Stadium has three main concession stands, located behind the south, north and west stands. The top-selling items are peanuts, hot dogs and soda.

More information

Review of Linear Algebra

Review of Linear Algebra Review of Linear Algebra Dr Gerhard Roth COMP 40A Winter 05 Version Linear algebra Is an important area of mathematics It is the basis of computer vision Is very widely taught, and there are many resources

More information

Computer Graphics: 2D Transformations. Course Website:

Computer Graphics: 2D Transformations. Course Website: Computer Graphics: D Transformations Course Website: http://www.comp.dit.ie/bmacnamee 5 Contents Wh transformations Transformations Translation Scaling Rotation Homogeneous coordinates Matri multiplications

More information

ICS 6N Computational Linear Algebra Matrix Algebra

ICS 6N Computational Linear Algebra Matrix Algebra ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix

More information

Phys 201. Matrices and Determinants

Phys 201. Matrices and Determinants Phys 201 Matrices and Determinants 1 1.1 Matrices 1.2 Operations of matrices 1.3 Types of matrices 1.4 Properties of matrices 1.5 Determinants 1.6 Inverse of a 3 3 matrix 2 1.1 Matrices A 2 3 7 =! " 1

More information

ES.1803 Topic 16 Notes Jeremy Orloff

ES.1803 Topic 16 Notes Jeremy Orloff ES803 Topic 6 Notes Jerem Orloff 6 Eigenalues, diagonalization, decoupling This note coers topics that will take us seeral classes to get through We will look almost eclusiel at 2 2 matrices These hae

More information

CS-184: Computer Graphics. Today

CS-184: Computer Graphics. Today CS-184: Computer Graphics Lecture #3: 2D Transformations Prof. James O Brien Universit of California, Berkele V2006-S-03-1.0 Toda 2D Transformations Primitive Operations Scale, Rotate, Shear, Flip, Translate

More information

CS 354R: Computer Game Technology

CS 354R: Computer Game Technology CS 354R: Computer Game Technolog Transformations Fall 207 Universit of Teas at Austin CS 354R Game Technolog S. Abraham Transformations What are the? Wh should we care? Universit of Teas at Austin CS 354R

More information

. a m1 a mn. a 1 a 2 a = a n

. a m1 a mn. a 1 a 2 a = a n Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by

More information

ENR202 Mechanics of Materials Lecture 12B Slides and Notes

ENR202 Mechanics of Materials Lecture 12B Slides and Notes ENR0 Mechanics of Materials Lecture 1B Slides and Notes Slide 1 Copright Notice Do not remove this notice. COMMMONWEALTH OF AUSTRALIA Copright Regulations 1969 WARNING This material has been produced and

More information

Dot Products, Transposes, and Orthogonal Projections

Dot Products, Transposes, and Orthogonal Projections Dot Products, Transposes, and Orthogonal Projections David Jekel November 13, 2015 Properties of Dot Products Recall that the dot product or standard inner product on R n is given by x y = x 1 y 1 + +

More information

Math 4377/6308 Advanced Linear Algebra

Math 4377/6308 Advanced Linear Algebra 2.3 Composition Math 4377/6308 Advanced Linear Algebra 2.3 Composition of Linear Transformations Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math4377

More information

Matrices: 2.1 Operations with Matrices

Matrices: 2.1 Operations with Matrices Goals In this chapter and section we study matrix operations: Define matrix addition Define multiplication of matrix by a scalar, to be called scalar multiplication. Define multiplication of two matrices,

More information

The Force Table Introduction: Theory:

The Force Table Introduction: Theory: 1 The Force Table Introduction: "The Force Table" is a simple tool for demonstrating Newton s First Law and the vector nature of forces. This tool is based on the principle of equilibrium. An object is

More information

Elementary maths for GMT

Elementary maths for GMT Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1

More information

Graphics (INFOGR), , Block IV, lecture 8 Deb Panja. Today: Matrices. Welcome

Graphics (INFOGR), , Block IV, lecture 8 Deb Panja. Today: Matrices. Welcome Graphics (INFOGR), 2017-18, Block IV, lecture 8 Deb Panja Today: Matrices Welcome 1 Today Matrices: why and what? Matrix operations Determinants Adjoint/adjugate and inverse of matrices Geometric interpretation

More information

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections )

10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections ) c Dr. Igor Zelenko, Fall 2017 1 10. Linear Systems of ODEs, Matrix multiplication, superposition principle (parts of sections 7.2-7.4) 1. When each of the functions F 1, F 2,..., F n in right-hand side

More information

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector.

LESSON 35: EIGENVALUES AND EIGENVECTORS APRIL 21, (1) We might also write v as v. Both notations refer to a vector. LESSON 5: EIGENVALUES AND EIGENVECTORS APRIL 2, 27 In this contet, a vector is a column matri E Note 2 v 2, v 4 5 6 () We might also write v as v Both notations refer to a vector (2) A vector can be man

More information

Section 1.6. M N = [a ij b ij ], (1.6.2)

Section 1.6. M N = [a ij b ij ], (1.6.2) The Calculus of Functions of Several Variables Section 16 Operations with Matrices In the previous section we saw the important connection between linear functions and matrices In this section we will

More information

Announcements Wednesday, October 10

Announcements Wednesday, October 10 Announcements Wednesday, October 10 The second midterm is on Friday, October 19 That is one week from this Friday The exam covers 35, 36, 37, 39, 41, 42, 43, 44 (through today s material) WeBWorK 42, 43

More information

( ) ( ) ( ) ( ) TNM046: Datorgrafik. Transformations. Linear Algebra. Linear Algebra. Sasan Gooran VT Transposition. Scalar (dot) product:

( ) ( ) ( ) ( ) TNM046: Datorgrafik. Transformations. Linear Algebra. Linear Algebra. Sasan Gooran VT Transposition. Scalar (dot) product: TNM046: Datorgrafik Transformations Sasan Gooran VT 04 Linear Algebra ( ) ( ) =,, 3 =,, 3 Transposition t = 3 t = 3 Scalar (dot) product: Length (Norm): = t = + + 3 3 = = + + 3 Normaliation: ˆ = Linear

More information

Announcements Monday, October 02

Announcements Monday, October 02 Announcements Monday, October 02 Please fill out the mid-semester survey under Quizzes on Canvas WeBWorK 18, 19 are due Wednesday at 11:59pm The quiz on Friday covers 17, 18, and 19 My office is Skiles

More information

Matrices BUSINESS MATHEMATICS

Matrices BUSINESS MATHEMATICS Matrices BUSINESS MATHEMATICS 1 CONTENTS Matrices Special matrices Operations with matrices Matrix multipication More operations with matrices Matrix transposition Symmetric matrices Old exam question

More information

Matrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation.

Matrices. Math 240 Calculus III. Wednesday, July 10, Summer 2013, Session II. Matrices. Math 240. Definitions and Notation. function Matrices Calculus III Summer 2013, Session II Wednesday, July 10, 2013 Agenda function 1. 2. function function Definition An m n matrix is a rectangular array of numbers arranged in m horizontal

More information

Game Engineering: 2D

Game Engineering: 2D Game Engineering: 2D CS420-2010F-06 2D Math David Galles Department of Computer Science University of San Francisco 06-0: Back to Basics A Vector is a displacement Vector has both direction and length

More information

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij

Topics. Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij Topics Vectors (column matrices): Vector addition and scalar multiplication The matrix of a linear function y Ax The elements of a matrix A : A ij or a ij lives in row i and column j Definition of a matrix

More information

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010

What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 What is the Matrix? Linear control of finite-dimensional spaces. November 28, 2010 Scott Strong sstrong@mines.edu Colorado School of Mines What is the Matrix? p. 1/20 Overview/Keywords/References Advanced

More information

Linear Algebra and Matrix Inversion

Linear Algebra and Matrix Inversion Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much

More information

Recitation 8: Graphs and Adjacency Matrices

Recitation 8: Graphs and Adjacency Matrices Math 1b TA: Padraic Bartlett Recitation 8: Graphs and Adjacency Matrices Week 8 Caltech 2011 1 Random Question Suppose you take a large triangle XY Z, and divide it up with straight line segments into

More information

Matrices. VCE Maths Methods - Unit 2 - Matrices

Matrices. VCE Maths Methods - Unit 2 - Matrices Matrices Introduction to matrices Addition & subtraction Scalar multiplication Matri multiplication The unit matri Matri division - the inverse matri Using matrices - simultaneous equations Matri transformations

More information

1 Matrices and matrix algebra

1 Matrices and matrix algebra 1 Matrices and matrix algebra 1.1 Examples of matrices A matrix is a rectangular array of numbers and/or variables. For instance 4 2 0 3 1 A = 5 1.2 0.7 x 3 π 3 4 6 27 is a matrix with 3 rows and 5 columns

More information

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2

We could express the left side as a sum of vectors and obtain the Vector Form of a Linear System: a 12 a x n. a m2 Week 22 Equations, Matrices and Transformations Coefficient Matrix and Vector Forms of a Linear System Suppose we have a system of m linear equations in n unknowns a 11 x 1 + a 12 x 2 + + a 1n x n b 1

More information

18.06 Problem Set 3 Due Wednesday, 27 February 2008 at 4 pm in

18.06 Problem Set 3 Due Wednesday, 27 February 2008 at 4 pm in 8.6 Problem Set 3 Due Wednesday, 27 February 28 at 4 pm in 2-6. Problem : Do problem 7 from section 2.7 (pg. 5) in the book. Solution (2+3+3+2 points) a) False. One example is when A = [ ] 2. 3 4 b) False.

More information

Vectors and the Geometry of Space

Vectors and the Geometry of Space Chapter 12 Vectors and the Geometr of Space Comments. What does multivariable mean in the name Multivariable Calculus? It means we stud functions that involve more than one variable in either the input

More information

Section 9.2: Matrices.. a m1 a m2 a mn

Section 9.2: Matrices.. a m1 a m2 a mn Section 9.2: Matrices Definition: A matrix is a rectangular array of numbers: a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn In general, a ij denotes the (i, j) entry of A. That is, the entry in

More information

Mathematics 13: Lecture 10

Mathematics 13: Lecture 10 Mathematics 13: Lecture 10 Matrices Dan Sloughter Furman University January 25, 2008 Dan Sloughter (Furman University) Mathematics 13: Lecture 10 January 25, 2008 1 / 19 Matrices Recall: A matrix is a

More information

the Further Mathematics network

the Further Mathematics network the Further Mathematics network www.fmnetwork.org.uk 1 the Further Mathematics network www.fmnetwork.org.uk Further Pure 3: Teaching Vector Geometry Let Maths take you Further 2 Overview Scalar and vector

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

Math 265 Midterm 2 Review

Math 265 Midterm 2 Review Math 65 Midterm Review March 6, 06 Things you should be able to do This list is not meant to be ehaustive, but to remind you of things I may ask you to do on the eam. These are roughly in the order they

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I ICS4: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by

More information

Math 314/814 Topics for first exam

Math 314/814 Topics for first exam Chapter 2: Systems of linear equations Math 314/814 Topics for first exam Some examples Systems of linear equations: 2x 3y z = 6 3x + 2y + z = 7 Goal: find simultaneous solutions: all x, y, z satisfying

More information

Matrix Arithmetic. j=1

Matrix Arithmetic. j=1 An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the i-th row and j-th column

More information

Now, if you see that x and y are present in both equations, you may write:

Now, if you see that x and y are present in both equations, you may write: Matrices: Suppose you have two simultaneous equations: y y 3 () Now, if you see that and y are present in both equations, you may write: y 3 () You should be able to see where the numbers have come from.

More information

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation

More information

Mathematics for Graphics and Vision

Mathematics for Graphics and Vision Mathematics for Graphics and Vision Steven Mills March 3, 06 Contents Introduction 5 Scalars 6. Visualising Scalars........................ 6. Operations on Scalars...................... 6.3 A Note on

More information

I = i 0,

I = i 0, Special Types of Matrices Certain matrices, such as the identity matrix 0 0 0 0 0 0 I = 0 0 0, 0 0 0 have a special shape, which endows the matrix with helpful properties The identity matrix is an example

More information

Basic Linear Algebra in MATLAB

Basic Linear Algebra in MATLAB Basic Linear Algebra in MATLAB 9.29 Optional Lecture 2 In the last optional lecture we learned the the basic type in MATLAB is a matrix of double precision floating point numbers. You learned a number

More information

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns.

Section 9.2: Matrices. Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. Section 9.2: Matrices Definition: A matrix A consists of a rectangular array of numbers, or elements, arranged in m rows and n columns. That is, a 11 a 12 a 1n a 21 a 22 a 2n A =...... a m1 a m2 a mn A

More information

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication.

Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. 7-6 Multiplying matrices by diagonal matrices is faster than usual matrix multiplication. The following equations generalize to matrices of any size. Multiplying a matrix from the left by a diagonal matrix

More information

Matrices. Chapter Definitions and Notations

Matrices. Chapter Definitions and Notations Chapter 3 Matrices 3. Definitions and Notations Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which

More information

MATH 1210 Assignment 4 Solutions 16R-T1

MATH 1210 Assignment 4 Solutions 16R-T1 MATH 1210 Assignment 4 Solutions 16R-T1 Attempt all questions and show all your work. Due November 13, 2015. 1. Prove using mathematical induction that for any n 2, and collection of n m m matrices A 1,

More information

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which?

Two conventions for coordinate systems. Left-Hand vs Right-Hand. x z. Which is which? walters@buffalo.edu CSE 480/580 Lecture 2 Slide 3-D Transformations 3-D space Two conventions for coordinate sstems Left-Hand vs Right-Hand (Thumb is the ais, inde is the ais) Which is which? Most graphics

More information

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations

Chapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2

More information

Lecture 3: Matrix and Matrix Operations

Lecture 3: Matrix and Matrix Operations Lecture 3: Matrix and Matrix Operations Representation, row vector, column vector, element of a matrix. Examples of matrix representations Tables and spreadsheets Scalar-Matrix operation: Scaling a matrix

More information

Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3. Diagonal matrices

Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3. Diagonal matrices Roberto s Notes on Linear Algebra Chapter 10: Eigenvalues and diagonalization Section 3 Diagonal matrices What you need to know already: Basic definition, properties and operations of matrix. What you

More information

Get Solution of These Packages & Learn by Video Tutorials on Matrices

Get Solution of These Packages & Learn by Video Tutorials on  Matrices FEE Download Stud Package from website: wwwtekoclassescom & wwwmathsbsuhagcom Get Solution of These Packages & Learn b Video Tutorials on wwwmathsbsuhagcom Matrices An rectangular arrangement of numbers

More information

Linear Algebraic Equations

Linear Algebraic Equations Linear Algebraic Equations Linear Equations: a + a + a + a +... + a = c 11 1 12 2 13 3 14 4 1n n 1 a + a + a + a +... + a = c 21 2 2 23 3 24 4 2n n 2 a + a + a + a +... + a = c 31 1 32 2 33 3 34 4 3n n

More information

Chapter 18 Quadratic Function 2

Chapter 18 Quadratic Function 2 Chapter 18 Quadratic Function Completed Square Form 1 Consider this special set of numbers - the square numbers or the set of perfect squares. 4 = = 9 = 3 = 16 = 4 = 5 = 5 = Numbers like 5, 11, 15 are

More information

MATRICES. a m,1 a m,n A =

MATRICES. a m,1 a m,n A = MATRICES Matrices are rectangular arrays of real or complex numbers With them, we define arithmetic operations that are generalizations of those for real and complex numbers The general form a matrix of

More information

1 Matrices and Systems of Linear Equations. a 1n a 2n

1 Matrices and Systems of Linear Equations. a 1n a 2n March 31, 2013 16-1 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real

More information

Matrix representation of a linear map

Matrix representation of a linear map Matrix representation of a linear map As before, let e i = (0,..., 0, 1, 0,..., 0) T, with 1 in the i th place and 0 elsewhere, be standard basis vectors. Given linear map f : R n R m we get n column vectors

More information

And as h is a column vector, recall that h t is the transpose of h, that is, the corresponding row vector, and so the computation becomes

And as h is a column vector, recall that h t is the transpose of h, that is, the corresponding row vector, and so the computation becomes The Hessian and optimization Let us start with two dimensions: Let f (x, ) be a function of two variables. We write the Talor expansion around (x, ). Write the vector h = x x,. We will consider h as a

More information

Matrix & Linear Algebra

Matrix & Linear Algebra Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A

More information

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds

Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds Linear Algebra Tutorial for Math3315/CSE3365 Daniel R. Reynolds These notes are meant to provide a brief introduction to the topics from Linear Algebra that will be useful in Math3315/CSE3365, Introduction

More information

Lecture 27: More on Rotational Kinematics

Lecture 27: More on Rotational Kinematics Lecture 27: More on Rotational Kinematics Let s work out the kinematics of rotational motion if α is constant: dω α = 1 2 α dω αt = ω ω ω = αt + ω ( t ) dφ α + ω = dφ t 2 α + ωo = φ φo = 1 2 = t o 2 φ

More information

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.

Lecture 5. Equations of Lines and Planes. Dan Nichols MATH 233, Spring 2018 University of Massachusetts. Lecture 5 Equations of Lines and Planes Dan Nichols nichols@math.umass.edu MATH 233, Spring 2018 Universit of Massachusetts Februar 6, 2018 (2) Upcoming midterm eam First midterm: Wednesda Feb. 21, 7:00-9:00

More information

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic

A FIRST COURSE IN LINEAR ALGEBRA. An Open Text by Ken Kuttler. Matrix Arithmetic A FIRST COURSE IN LINEAR ALGEBRA An Open Text by Ken Kuttler Matrix Arithmetic Lecture Notes by Karen Seyffarth Adapted by LYRYX SERVICE COURSE SOLUTION Attribution-NonCommercial-ShareAlike (CC BY-NC-SA)

More information

Sometimes the domains X and Z will be the same, so this might be written:

Sometimes the domains X and Z will be the same, so this might be written: II. MULTIVARIATE CALCULUS The first lecture covered functions where a single input goes in, and a single output comes out. Most economic applications aren t so simple. In most cases, a number of variables

More information

A primer on matrices

A primer on matrices A primer on matrices Stephen Boyd August 4, 2007 These notes describe the notation of matrices, the mechanics of matrix manipulation, and how to use matrices to formulate and solve sets of simultaneous

More information

Math 54 Homework 3 Solutions 9/

Math 54 Homework 3 Solutions 9/ Math 54 Homework 3 Solutions 9/4.8.8.2 0 0 3 3 0 0 3 6 2 9 3 0 0 3 0 0 3 a a/3 0 0 3 b b/3. c c/3 0 0 3.8.8 The number of rows of a matrix is the size (dimension) of the space it maps to; the number of

More information

Matrix Representation

Matrix Representation Matrix Representation Matrix Rep. Same basics as introduced already. Convenient method of working with vectors. Superposition Complete set of vectors can be used to express any other vector. Complete set

More information

Transformations. Chapter D Transformations Translation

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

More information

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p

Math Bootcamp An p-dimensional vector is p numbers put together. Written as. x 1 x =. x p Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

More information

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices

Matrices Gaussian elimination Determinants. Graphics 2009/2010, period 1. Lecture 4: matrices Graphics 2009/2010, period 1 Lecture 4 Matrices m n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in

More information

Foundations of Cryptography

Foundations of Cryptography Foundations of Cryptography Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi Department of Mathematics and Statistics University of Turku 2017 Ville Junnila, Arto Lepistö viljun@utu.fi, alepisto@utu.fi

More information

Review from Bootcamp: Linear Algebra

Review from Bootcamp: Linear Algebra Review from Bootcamp: Linear Algebra D. Alex Hughes October 27, 2014 1 Properties of Estimators 2 Linear Algebra Addition and Subtraction Transpose Multiplication Cross Product Trace 3 Special Matrices

More information

Analytic Trigonometry

Analytic Trigonometry CHAPTER 5 Analtic Trigonometr 5. Fundamental Identities 5. Proving Trigonometric Identities 5.3 Sum and Difference Identities 5.4 Multiple-Angle Identities 5.5 The Law of Sines 5.6 The Law of Cosines It

More information

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes

Mathematics 309 Conic sections and their applicationsn. Chapter 2. Quadric figures. ai,j x i x j + b i x i + c =0. 1. Coordinate changes Mathematics 309 Conic sections and their applicationsn Chapter 2. Quadric figures In this chapter want to outline quickl how to decide what figure associated in 2D and 3D to quadratic equations look like.

More information

M. Matrices and Linear Algebra

M. Matrices and Linear Algebra M. Matrices and Linear Algebra. Matrix algebra. In section D we calculated the determinants of square arrays of numbers. Such arrays are important in mathematics and its applications; they are called matrices.

More information

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full

n n matrices The system of m linear equations in n variables x 1, x 2,..., x n can be written as a matrix equation by Ax = b, or in full n n matrices Matrices Definitions Diagonal, Identity, and zero matrices Addition Multiplication Transpose and inverse The system of m linear equations in n variables x 1, x 2,..., x n a 11 x 1 + a 12 x

More information

ME751 Advanced Computational Multibody Dynamics

ME751 Advanced Computational Multibody Dynamics ME751 Advanced Computational Multibody Dynamics Review: Elements of Linear Algebra & Calculus September 9, 2016 Dan Negrut University of Wisconsin-Madison Quote of the day If you can't convince them, confuse

More information

Quick Tour of Linear Algebra and Graph Theory

Quick Tour of Linear Algebra and Graph Theory Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and

More information

k is a product of elementary matrices.

k is a product of elementary matrices. Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation

More information

MATRICES AND MATRIX OPERATIONS

MATRICES AND MATRIX OPERATIONS SIZE OF THE MATRIX is defined by number of rows and columns in the matrix. For the matrix that have m rows and n columns we say the size of the matrix is m x n. If matrix have the same number of rows (n)

More information

Comparing linear and exponential growth

Comparing linear and exponential growth Januar 16, 2009 Comparing Linear and Exponential Growth page 1 Comparing linear and exponential growth How does exponential growth, which we ve been studing this week, compare to linear growth, which we

More information

Graphics Example: Type Setting

Graphics Example: Type Setting D Transformations Graphics Eample: Tpe Setting Modern Computerized Tpesetting Each letter is defined in its own coordinate sstem And positioned on the page coordinate sstem It is ver simple, m she thought,

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

Ross Program 2017 Application Problems

Ross Program 2017 Application Problems Ross Program 2017 Application Problems This document is part of the application to the Ross Mathematics Program, and is posted at http://u.osu.edu/rossmath/. The Admission Committee will start reading

More information

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions

Math 308 Midterm Answers and Comments July 18, Part A. Short answer questions Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to

More information

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 50 - CHAPTER 3: MATRICES QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 03 marks Matrix A matrix is an ordered rectangular array of numbers

More information

Systems of Linear Equations and Matrices

Systems of Linear Equations and Matrices Chapter 1 Systems of Linear Equations and Matrices System of linear algebraic equations and their solution constitute one of the major topics studied in the course known as linear algebra. In the first

More information

A Tutorial on Euler Angles and Quaternions

A Tutorial on Euler Angles and Quaternions A Tutorial on Euler Angles and Quaternions Moti Ben-Ari Department of Science Teaching Weimann Institute of Science http://www.weimann.ac.il/sci-tea/benari/ Version.0.1 c 01 17 b Moti Ben-Ari. This work

More information

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform

we must pay attention to the role of the coordinate system w.r.t. which we perform a tform linear SO... we will want to represent the geometr of points in space we will often want to perform (rigid) transformations to these objects to position them translate rotate or move them in an animation

More information

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop

Fall Inverse of a matrix. Institute: UC San Diego. Authors: Alexander Knop Fall 2017 Inverse of a matrix Authors: Alexander Knop Institute: UC San Diego Row-Column Rule If the product AB is defined, then the entry in row i and column j of AB is the sum of the products of corresponding

More information

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E

Exercise Set Suppose that A, B, C, D, and E are matrices with the following sizes: A B C D E Determine the size of a given matrix. Identify the row vectors and column vectors of a given matrix. Perform the arithmetic operations of matrix addition, subtraction, scalar multiplication, and multiplication.

More information