# Linear transformations Affine transformations Transformations in 3D. Graphics 2011/2012, 4th quarter. Lecture 5: linear and affine transformations

Save this PDF as:

Size: px
Start display at page:

Download "Linear transformations Affine transformations Transformations in 3D. Graphics 2011/2012, 4th quarter. Lecture 5: linear and affine transformations"

## Transcription

1 Lecture 5 Linear and affine transformations

2 Vector transformation: basic idea Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiplication of an n n matrix with a vector (i.e. a n 1 matrix): ( a11 a In 2D: 12 a 21 a 22 ) ( ) x = y ( ) a11 x + a 12 y a 21 x + a 22 y a 11 a 12 a 13 x a 11 x + a 12 y + a 13 z In 3D: a 21 a 22 a 23 y = a 21 x + a 22 y + a 23 z a 31 a 32 a 33 z a 31 x + a 32 y + a 33 z The result is a (transformed) vector.

3 Example: scaling Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors To scale with a factor two with respect to the origin, we apply the matrix ( ) to a vector: ( ) ( ) 2 0 x 0 2 y = ( ) 2x + 0y = 0x + 2y ( ) 2x 2y

4 Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors A function T : R n R m is called a linear transformation if it satisfies 1 T ( u + v) = T ( u) + T ( v) for all u, v R n. 2 T (c v) = ct ( v) for all v R n and all scalars c. Or (alternatively), if it satisfies T (c 1 u + c 2 v) = c 1 T ( u) + c 2 T ( v) for all u, v R n and all scalars c 1, c 2. v T (v) u + v u T (u + v) T (u) Linear transformations can be represented by matrices

5 Example: scaling Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors We already saw scaling by a factor of 2. In general, a matrix ( a a 22 ) scales the i-th coordinate of a vector by the factor a ii 0, i.e. ( a a 22 ) ( ) x = y ( ) a11 x + 0y = 0x + a 22 y ( ) a11 x a 22 y

6 Example: scaling Linear transformations Scaling does not have to be uniform. Here, we scale with a factor one half in x-direction, and three in y-direction: ( 1 ) ( ) 2 0 x 0 3 y = ( 1 2 x ) 3y Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

7 Example: scaling Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Using Gaussian elimination to calculate the inverse of a matrix: ( ) (Gaussian elimination) ( )

8 Example: projection Linear transformations We can also use matrices to do orthographic projections, for instance, onto the Y -axis: ( ) ( ) 0 0 x 0 1 y = ( ) 0 y Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

9 Example: reflection Linear transformations Reflection in the line y = x boils down to swapping x- and y-coordinates: ( ) ( ) 0 1 x 1 0 y = ( ) y x Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Q: what is the inverse of this matrix?

10 Example: shearing Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Shearing in x-direction pushes things sideways : ( ) ( ) 1 1 x 0 1 y = ( ) x + y y We can also push things upwards with ( ) ( ) 1 0 x 1 1 y = ( x ) x + y

11 Example: shearing Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors General case for shearing in x-direction: ( ) ( ) 1 s x 0 1 y ( ) x + sy = y And its inverse operation: ( ) ( ) 1 s x 0 1 y ( ) x sy = y

12 Example: rotation Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors To rotate 45 about the origin, we apply the matrix ( ) = 2 2 ( ) Note: 2 2 = cos 45 = sin 45, so this is the same as ( cos 45 sin 45 ) sin 45 cos 45

13 Example: rotation Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors General case for a counterclockwise rotation about an angle φ around the origin: ( ) cos φ sin φ sin φ cos φ And for clockwise rotation: ( ) cos φ sin φ sin φ cos φ

14 Finding matrices Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Applying matrices is pretty straightforward, but how do we find the matrix for a given linear transformation? ( ) a11 a Let A = 12 a 21 a 22 Q: what values do we have to assign to the a ij s to achieve a certain transformation?

15 Finding matrices Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Let s apply some transformations to the base vectors b 1 = (1, 0) and b 2 = (0, 1) of the carthesian coordinates system, e.g. Scaling (factors a, b 0): Shearing (x-dir., factor s): Reflection (in line y = x): Rotation (countercl., 45 ): ( ) ( ) a 0 x 0 b y ( ) ( ) 1 s x 0 1 y ( ) ( ) 0 1 x 1 0 y 2 2 = = = ( ) ax by ( ) x + sy y ( ) y x ( ) ( ) 1 1 x 1 1 y = 2 2 ( ) x y x + y

16 Finding matrices Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Aha! The column vectors of a transformation matrix are the images of the base vectors! That gives us an easy method of finding the matrix for a given linear transformation. Why? Because matrix multiplication is a linear transformation.

17 Finding matrices Linear transformations Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Remember: T is a linear transformation if and only if T (c 1 u + c 2 v) = c 1 T ( u) + c 2 T ( v) Let s look at carthesian coordinates, where each vector w can be represented as a linear combination of the base vectors b 1, b 2 : w = ( ) x = x y ( ) 1 + y 0 ( ) 0 1 If we apply a linear transformation T to this vector, we get: ( ) ( ) ( ) ( ) ( ) x T ( ) = T (x + y ) = xt ( ) + yt ( ) y

18 Finding matrices: example Rotation (counterclockwise, angle φ): Definition Examples Finding matrices Compositions of transformations Transposing normal vectors ( cos φ ) sin φ sin φ cos φ For example for the first base vector b 1 :

19 Example: reflection and scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Multiple transformations can be combined into one. Here, we first do a reflection in the line y = x, and then we scale with a factor 5 in x-direction, and a factor 2 in y-direction: ( ) ( ) ( ) x = y ( ) ( ) x y Remember: matrix multiplication is associative, i.e. A(Bx) = (AB)x.

20 Example: reflection and scaling Definition Examples Finding matrices Compositions of transformations Transposing normal vectors But: Matrix multiplication is not commutative, i.e. in general AB BA, for example: ( ) ( ) ( ) x = y ( ) ( ) ( ) x = y Mind the order! ( ( ) ( ) x = y ) ( ) x = y ( ) 5y 2x ( ) 2y 5x

21 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Unfortunately, normal vectors are not always transformed properly. E.g. look at shearing, where tangent vectors are correctly transformed but normal vectors not. To transform a normal vector n correctly under a given linear transformation A, we have to apply the matrix (A 1 ) T. Why?

22 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors We know that tangent vectors are tranformed correctly: A t = t A. But this is not necessarily true for normal vectors: A n n A. Goal: find matrix N A that transforms n correctly, i.e. N A n = n N where n N is the correct normal vector of the transformed surface. We know that n T t = 0. Hence, we can also say that n T I t = 0 which is is the same as n T A 1 A t = 0

23 Transposing normal vectors Definition Examples Finding matrices Compositions of transformations Transposing normal vectors Because A t = t A is our correctly transformed tangent vector, we have n T A 1 t A = 0 Because their scaler product is 0, n T A 1 must be orthogonal to it. So, the vector we are looking for must be n T N = nt A 1. Because of how matrix multiplication is defined, this is a transposed vector. But we can rewrite this to n N = ( n T A 1 ) T. And if you remember that (AB) T = B T A T, we get n N = (A 1 ) T n

24 Problem and solution: homogeneous coordinates More complex transformations So now we know how to determine matrices for a given transformation. Let s try another one: Q: what is the matrix for a rotation of 90 about the point (2, 1)?

25 Problem and solution: homogeneous coordinates More complex transformations We can form our transformation by composing three simpler transformations: Translate everything such that the center of rotation maps to the origin. Rotate about the origin. Revert the translation from the first step. Q: but what is the matrix for a translation?

26 Problem and solution: homogeneous coordinates More complex transformations Translation is not a linear transformation. With linear transformations we get: ( ) ( ) a11 a Ax = 12 x = a 21 a 22 y But we need something like: ( x + ) xt y + y t ( a11 x + ) a 12 y a 21 x + a 22 y We can do this with a combination of linear transformations and translations called affine transformations.

27 Problem and solution: homogeneous coordinates Homogeneous coordinates Observation: Shearing in 2D smells a lot like translation in 1D...

28 Problem and solution: homogeneous coordinates Homogeneous coordinates... and shearing in 3D smells like translation in 2D (... and shearing in 4D... )

29 Problem and solution: homogeneous coordinates Homogeneous coordinates in 2D: basic idea

30 Problem and solution: homogeneous coordinates Homogeneous coordinates in 2D: basic idea We see: by adding a 3rd dimension to our 2D space, we can add a constant value to the first two coordinates by matrix multiplication. x x + x t M y = y + y t d d That s exactly what we want (for the first 2 coordinates). But how does the matrix M look like? And how are we dealing with this 3rd coordinate?

31 Problem and solution: homogeneous coordinates Homogeneous coordinates Shearing in 3D based on the z-coordinate is a simple generalization of 2D shearing: 1 0 x t x x + x t d 0 1 y t y = y + y t d d d Notice that we didn t make any assumption about d,...

32 Problem and solution: homogeneous coordinates Homogeneous coordinates: matrices... so we can use, for example, d = 1: 1 0 x t x x + x t 0 1 y t y = y + y t

33 Problem and solution: homogeneous coordinates Homogeneous coordinates: points Translations in 2D can be represented as shearing in 3D by looking at the plane z = 1. By representing all our 2D points (x, y) by 3D vectors (x, y, 1), we can translate them about (x t, y t ) using the following 3D shearing matrix: 1 0 x t x x + x t 0 1 y t y = y + y t

34 Problem and solution: homogeneous coordinates Homogeneous coordinates: vectors But can we translate a vector? No! (Remember: vectors are defined by their length and direction.) Hence, we have to represent a vector differently, i.e. by (x, y, 0): 1 0 x t x x 0 1 y t y = y

35 Problem and solution: homogeneous coordinates Homogeneous coordinates (i.e. linear transformations and translations) can be done with simple matrix multiplication if we add homogeneous coordinates, i.e. (in 2D): a third coordinate z = 1 x to each location: y 1 a third coordinate z = 0 x to each vector: y 0 a third row ( ) to each matrix:

36 Problem and solution: homogeneous coordinates : points vs. vectors Scaling and moving a location/point (or an object ): 1 0 x t s x s 1 0 x t x s 1x + x t 0 1 y t 0 s 2 0 y = 0 s 2 y t y = s 2y + y t With the same matrix, we scale (but not move!) a vector: s 1 0 x t x s 1x 0 s 2 y t y = s 2y

37 Problem and solution: homogeneous coordinates : example What is the matrix for reflection in the line y = x + 5? Idea: move everything to the origin, reflect, and the move everything back.

38 Problem and solution: homogeneous coordinates : example 1. Translation of y = x + 5 to y = x 1 0 x t x x + x t 0 1 y t y = y + y t Reflection on y = x: ( 0 ) Translate y = x back to y = x

39 Problem and solution: homogeneous coordinates So, the matrix for reflection in the line y = x + 5 is But what is the significance of the columns of the matrix?

40 Problem and solution: homogeneous coordinates The last column vector is the image of the origin after applying the affine transformation (Remember: The other columns are the image of the base vectors under the linear transformation) (Also remember: The coordinates in the last row are called homogeneous coordinates)

41 Moving up a dimension Rotations in 3D Reflections in 3D Linear transformations in 3D In general, linear transformations in 3D are a straightforward extension of their 2D counterpart. We have already seen an example: shearing in x y direction 1 0 d x 0 1 d y Or in general 1 d x2 d x3 d y1 1 d y

42 Moving up a dimension Rotations in 3D Reflections in 3D in 3D Homogeneous coordinates (and therewith affine transformations) in 3D are also a straightforward generalization from 2D. We just have to use 4 4 matrices now: a 11 a 12 a 13 0 Matrices: a 21 a 22 a 23 0 a 31 a 32 a x x points: y z and vectors: y z 1 0

43 Moving up a dimension Rotations in 3D Reflections in 3D For scaling, we have three scaling factors on the diagonal of the matrix. s x s y s z Shearing can be done in either x-, y-, or z-direction (or a combination thereof). For example, shearing in x-direction: 1 d x2 d x3 x x + d y y + d z z y = y z z

44 Moving up a dimension Rotations in 3D Reflections in 3D We see: transformations in 3D are very similar to those in 2D. Scaling and shearing are straightforward generalizations of the 2D cases. Reflection is done with respect to planes Rotation is done about directed lines.

45 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Rotation in 2D (x y-plane): ( ) cos φ sin φ sin φ cos φ

46 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the z-axis? Rotation in 2D (x y-plane): ( cos φ ) sin φ sin φ cos φ Rotation in 3D around z-axis: cos φ sin φ 0 sin φ cos φ

47 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Q: What is the matrix for a rotation of angle φ about the directed line (0, 1, 2) + t(3, 0, 4) t > 0?

48 Moving up a dimension Rotations in 3D Reflections in 3D : rotations We need a 3D transformation that rotates around an arbitrary vector w. How can we do that? Idea: 1 Rotate vector w to z-axis 2 Do 3D rotation around z-axis 3 Rotate everything back to original position

49 Moving up a dimension Rotations in 3D Reflections in 3D : rotations We already know how to rotate around the z-axis: cos φ sin φ 0 sin φ cos φ

50 Moving up a dimension Rotations in 3D Reflections in 3D : rotations But how do we get the other 2 rotation matrices? cos φ sin φ 0? sin φ cos φ 0? Rotating everything back to it s original position after we have done the rotation is easy. Remember that the columns of the rotation matrix represent the images of the carthesian basis vectors under the transformation! x u x v x w cos φ sin φ 0 y u y v y w sin φ cos φ 0? z u z v z w 0 0 1

51 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Now all we still need is the original rotation matrix x u x v x w cos φ sin φ 0 y u y v y w sin φ cos φ 0? z u z v z w But that s just the inverse of the other rotation matrix. Note: if we assume that u = v = w = 1 we don t even need to calculate it, because the inverse of an orthogonal matrix is always its transposed! x u x v x w cos φ sin φ 0 x u y u z u y u y v y w sin φ cos φ 0 x v y v z v z u z v z w x w y w z w

52 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Hmm, but we only have the vector w. How do we get a whole coordinates system u, v, w? w

53 Moving up a dimension Rotations in 3D Reflections in 3D : rotations Notice that such a rotation is not unique. (And it doesn t have to be as long as the two rotation matrices are correct.)

54 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? What happens to a random point (x, y, z) in 3D when we reflect it on the XY -plane?

55 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Hence, the matrix for reflection in XY -plane is:

56 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 0?

57 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 0?

58 Moving up a dimension Rotations in 3D Reflections in 3D : reflections Q: What is the matrix for reflection in XY -plane? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 0? Q: What is the matrix for reflection in the plane 3x + 4y 12z = 11?

### Algebra and Linear Algebra

Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...

### 2D Geometric Transformations. COMP 770 Fall 2011

2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication

### Motivation. Goals. General Idea. Outline. (Nonuniform) Scale. Foundations of Computer Graphics

Foundations of Computer Graphics Basic 2D Transforms Motivation Many different coordinate systems in graphics World, model, body, arms, To relate them, we must transform between them Also, for modeling

### 0.1 Linear Transformations

.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called

### geometric transforms

geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition

### 3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)

3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis

### Vector Notation: AB represents the vector from point A to point B on a graph. The vector can be computed by B A.

1 Linear Transformations Prepared by: Robin Michelle King A transformation of an object is a change in position or dimension (or both) of the object. The resulting object after the transformation is called

### 6 Systems of Linear Equations

6 Systems of Linear Equations A system of equations of the form or is called a linear system of equations. x + y = 7 x y = 5p 6q + r = p + q 5r = 7 6p q + r = Definition. An equation involving certain

### 1.5 Elementary Matrices and a Method for Finding the Inverse

.5 Elementary Matrices and a Method for Finding the Inverse Definition A n n matrix is called an elementary matrix if it can be obtained from I n by performing a single elementary row operation Reminder:

### Introduction to Matrices

Introduction to Matrices Tom Davis tomrdavis@earthlinknet 1 Definitions A matrix (plural: matrices) is simply a rectangular array of things For now, we ll assume the things are numbers, but as you go on

### Math 54. Selected Solutions for Week Is u in the plane in R 3 spanned by the columns

Math 5. Selected Solutions for Week 2 Section. (Page 2). Let u = and A = 5 2 6. Is u in the plane in R spanned by the columns of A? (See the figure omitted].) Why or why not? First of all, the plane in

### 5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES

5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL MATRICES Definition 5.3. Orthogonal transformations and orthogonal matrices A linear transformation T from R n to R n is called orthogonal if it preserves

### Lecture 23: The Inverse of a Matrix

Lecture 23: The Inverse of a Matrix Winfried Just, Ohio University March 9, 2016 The definition of the matrix inverse Let A be an n n square matrix. The inverse of A is an n n matrix A 1 such that A 1

### Lecture 21: The Inverse of a Matrix

Lecture 21: The Inverse of a Matrix Winfried Just, Ohio University October 16, 2015 Review: Our chemical reaction system Recall our chemical reaction system A + 2B 2C A + B D A + 2C 2D B + D 2C If we write

### Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

### Geometric Transformation CS 211A

Geometric Transformation CS 211A What is transformation? Moving points (x,y) moves to (x+t, y+t) Can be in any dimension 2D Image warps 3D 3D Graphics and Vision Can also be considered as a movement to

### Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.

MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a 11

### 9 MATRICES AND TRANSFORMATIONS

9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the

### Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional

### 13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

### Rotation Matrices and Homogeneous Transformations

Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n

### 9 Matrices, determinants, inverse matrix, Cramer s Rule

AAC - Business Mathematics I Lecture #9, December 15, 2007 Katarína Kálovcová 9 Matrices, determinants, inverse matrix, Cramer s Rule Basic properties of matrices: Example: Addition properties: Associative:

### 2D Geometric Transformations

2D Geometric Transformations (Chapter 5 in FVD) 2D Geometric Transformations Question: How do we represent a geometric object in the plane? Answer: For now, assume that objects consist of points and lines.

### Orthogonal Matrices. u v = u v cos(θ) T (u) + T (v) = T (u + v). It s even easier to. If u and v are nonzero vectors then

Part 2. 1 Part 2. Orthogonal Matrices If u and v are nonzero vectors then u v = u v cos(θ) is 0 if and only if cos(θ) = 0, i.e., θ = 90. Hence, we say that two vectors u and v are perpendicular or orthogonal

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### Section 2.1. Section 2.2. Exercise 6: We have to compute the product AB in two ways, where , B =. 2 1 3 5 A =

Section 2.1 Exercise 6: We have to compute the product AB in two ways, where 4 2 A = 3 0 1 3, B =. 2 1 3 5 Solution 1. Let b 1 = (1, 2) and b 2 = (3, 1) be the columns of B. Then Ab 1 = (0, 3, 13) and

### We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### Matrices in Statics and Mechanics

Matrices in Statics and Mechanics Casey Pearson 3/19/2012 Abstract The goal of this project is to show how linear algebra can be used to solve complex, multi-variable statics problems as well as illustrate

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### MAT188H1S Lec0101 Burbulla

Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u

### Math E-301: Homework 4 Notes Due 10/5/09

Math E-301: Homework 4 Notes Due 10/5/09 The notes for homeworks will vary. For some problems, I may give one solution, but there may be other ways to approach the problem. I will try to mention these

### Lecture 2 Matrix Operations

Lecture 2 Matrix Operations transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2 1 Matrix transpose transpose of m n matrix A, denoted A T or

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### Matrices Summary. To add or subtract matrices they must be the same dimensions. Just add or subtract the corresponding numbers.

Matrices Summary To transpose a matrix write the rows as columns. Academic Skills Advice For example: 2 1 A = [ 1 2 1 0 0 9] A T = 4 2 2 1 2 1 1 0 4 0 9 2 To add or subtract matrices they must be the same

### Linear Algebra Review. Vectors

Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length

### Section V.3: Dot Product

Section V.3: Dot Product Introduction So far we have looked at operations on a single vector. There are a number of ways to combine two vectors. Vector addition and subtraction will not be covered here,

### Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday.

Math 312, Fall 2012 Jerry L. Kazdan Problem Set 5 Due: In class Thursday, Oct. 18 Late papers will be accepted until 1:00 PM Friday. In addition to the problems below, you should also know how to solve

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Row and column operations

Row and column operations It is often very useful to apply row and column operations to a matrix. Let us list what operations we re going to be using. 3 We ll illustrate these using the example matrix

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### Matrix Algebra LECTURE 1. Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 = a 11 x 1 + a 12 x 2 + +a 1n x n,

LECTURE 1 Matrix Algebra Simultaneous Equations Consider a system of m linear equations in n unknowns: y 1 a 11 x 1 + a 12 x 2 + +a 1n x n, (1) y 2 a 21 x 1 + a 22 x 2 + +a 2n x n, y m a m1 x 1 +a m2 x

### Chapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors

Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.

Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product

### Geometric Transformations

CS3 INTRODUCTION TO COMPUTER GRAPHICS Geometric Transformations D and 3D CS3 INTRODUCTION TO COMPUTER GRAPHICS Grading Plan to be out Wednesdas one week after the due date CS3 INTRODUCTION TO COMPUTER

### Diagonal, Symmetric and Triangular Matrices

Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by

### Chapter 6. Linear Transformation. 6.1 Intro. to Linear Transformation

Chapter 6 Linear Transformation 6 Intro to Linear Transformation Homework: Textbook, 6 Ex, 5, 9,, 5,, 7, 9,5, 55, 57, 6(a,b), 6; page 7- In this section, we discuss linear transformations 89 9 CHAPTER

### Matrix Inverse and Determinants

DM554 Linear and Integer Programming Lecture 5 and Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark Outline 1 2 3 4 and Cramer s rule 2 Outline 1 2 3 4 and

### Affine Transformations

Affine Transformations Reading Foley et al., Chapter 5.6 and Chapter 6 Supplemental David F. Rogers and J. Alan Adams, Mathematical Elements for Computer Graphics, Second edition 2D geometry Pipeline 3D

### 1 Scalars, Vectors and Tensors

DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical

### B such that AB = I and BA = I. (We say B is an inverse of A.) Definition A square matrix A is invertible (or nonsingular) if matrix

Matrix inverses Recall... Definition A square matrix A is invertible (or nonsingular) if matrix B such that AB = and BA =. (We say B is an inverse of A.) Remark Not all square matrices are invertible.

### Linear Equations in Linear Algebra

1 Linear Equations in Linear Algebra 1.8 INTRODUCTION TO LINEAR TRANSFORMATIONS LINEAR TRANSFORMATIONS A transformation (or function or mapping) T from R n to R m is a rule that assigns to each vector

### Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

### Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

### A general element of R 3 3 has the form A = d e f and the condition A T = A becomes. b e f, so that. b = d, c = g, and f = h. Thus, A = a b c.

Let V = {f(t) P 3 f() = f()} Show that V is a subspace of P 3 and find a basis of V The neutral element of P 3 is the function n(t) = In particular, n() = = n(), so n(t) V Let f, g V Then f() = f() and

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

### TWO-DIMENSIONAL TRANSFORMATION

CHAPTER 2 TWO-DIMENSIONAL TRANSFORMATION 2.1 Introduction As stated earlier, Computer Aided Design consists of three components, namely, Design (Geometric Modeling), Analysis (FEA, etc), and Visualization

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### Lecture 3: Coordinate Systems and Transformations

Lecture 3: Coordinate Systems and Transformations Topics: 1. Coordinate systems and frames 2. Change of frames 3. Affine transformations 4. Rotation, translation, scaling, and shear 5. Rotation about an

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain

Lectures notes on orthogonal matrices (with exercises) 92.222 - Linear Algebra II - Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n real-valued matrix A is said to be an orthogonal

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Numerical Methods Lecture 2 Simultaneous Equations

Numerical Methods Lecture 2 Simultaneous Equations Topics: matrix operations solving systems of equations Matrix operations: Mathcad is designed to be a tool for quick and easy manipulation of matrix forms

### Introduction to Matrices for Engineers

Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

### Chapter 4: Binary Operations and Relations

c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

### L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014

L1-2. Special Matrix Operations: Permutations, Transpose, Inverse, Augmentation 12 Aug 2014 Unfortunately, no one can be told what the Matrix is. You have to see it for yourself. -- Morpheus Primary concepts:

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: b k

MATRIX ALGEBRA FOR STATISTICS: PART 1 Matrices provide a compact notation for expressing systems of equations or variables. For instance, a linear function might be written as: y = x 1 b 1 + x 2 + x 3

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Systems of Linear Equations and Matrices

Chapter 2 Systems of Linear Equations and Matrices 2.1 Systems of Linear Equations - Introduction We have already solved a small system of linear equations when we found the intersection point of two lines.

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### Some linear transformations on R 2 Math 130 Linear Algebra D Joyce, Fall 2015

Some linear transformations on R 2 Math 3 Linear Algebra D Joce, Fall 25 Let s look at some some linear transformations on the plane R 2. We ll look at several kinds of operators on R 2 including reflections,

### Affine Transformations

A P P E N D I X C Affine Transformations CONTENTS C The need for geometric transformations 335 C2 Affine transformations 336 C3 Matri representation of the linear transformations 338 C4 Homogeneous coordinates

### Linear Algebra Notes

Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### Lecture 5: Singular Value Decomposition SVD (1)

EEM3L1: Numerical and Analytical Techniques Lecture 5: Singular Value Decomposition SVD (1) EE3L1, slide 1, Version 4: 25-Sep-02 Motivation for SVD (1) SVD = Singular Value Decomposition Consider the system

### Lecture 2: Homogeneous Coordinates, Lines and Conics

Lecture 2: Homogeneous Coordinates, Lines and Conics 1 Homogeneous Coordinates In Lecture 1 we derived the camera equations λx = P X, (1) where x = (x 1, x 2, 1), X = (X 1, X 2, X 3, 1) and P is a 3 4

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

### Matrix Calculations: Kernels & Images, Matrix Multiplication

Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers) Institute for Computing and Information Sciences Intelligent Systems Version: spring 2016 A. Kissinger Version:

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### Rotation in the Space

Rotation in the Space (Com S 477/577 Notes) Yan-Bin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One

### The calibration problem was discussed in details during lecture 3.

1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference

### Section 5.3. Section 5.3. u m ] l jj. = l jj u j + + l mj u m. v j = [ u 1 u j. l mj

Section 5. l j v j = [ u u j u m ] l jj = l jj u j + + l mj u m. l mj Section 5. 5.. Not orthogonal, the column vectors fail to be perpendicular to each other. 5..2 his matrix is orthogonal. Check that

### Representations of Molecular Properties

Representations of Molecular Properties We will routinely use sets of vectors located on each atom of a molecule to represent certain molecular properties of interest (e.g., orbitals, vibrations). The

### Notes from FE Review for mathematics, University of Kentucky

Notes from FE Review for mathematics, University of Kentucky Dr. Alan Demlow March 19, 2012 Introduction These notes are based on reviews for the mathematics portion of the Fundamentals of Engineering

### Lecture No. # 02 Prologue-Part 2

Advanced Matrix Theory and Linear Algebra for Engineers Prof. R.Vittal Rao Center for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture No. # 02 Prologue-Part 2 In the last