Fundamentals of Matrix Algebra, 2011a

More documents

Recommendations

Info

Chapter 5 Graphical Exploraons of Vectors Right now we are focusing on transforming the Cartesian plane – we are making 2D transformaons. Knowing how to do this provides a foundaon for transforming 3D space, 7 which, among other things, is very important when producing 3D computer graphics. Basic shapes can be drawn and then rotated, stretched, and/or moved to other regions of space. This also allows for things like “moving the camera view.” What kinds of transformaons are possible? We have already seen some of the things that are possible: rotaons, stretches, and flips. We have also menoned some things that are not possible. For instance, we stated that straight lines always get transformed to straight lines. Therefore, we cannot transform the unit square into a circle using a matrix. Let’s look at some common transformaons of the Cartesian plane and the matrices that perform these operaons. In the following figures, a transformaon matrix will be given alongside a picture of the transformed unit square. (The original unit square is drawn lightly as well to serve as a reference.) 2D Matrix Transformaons y Horizontal stretch by a factor of k. [ ] k 0 0 1 (k, 1) . x y Vercal stretch by a factor of k. [ ] 1 0 0 k (1, k) . x 7 Actually, it provides a foundaon for doing it in 4D, 5D, . . ., 17D, etc. Those are just harder to visualize. 196
5.1 Transformaons of the Cartesian Plane y Horizontal shear by a factor of k. [ ] 1 k 0 1 (k, 1) . x y Vercal shear by a factor of k. [ ] 1 0 k 1 (k, 1) . x Horizontal reflecon across the y axis. [ −1 ] 0 . x 0 1 y Vercal reflecon across the x axis. [ 1 0 ] . x 0 −1 y 197
Page 1:
. . . Fundamentals of Matrix Algebr
Page 4 and 5:
Copyright © 2011 Gregory Hartman L
Page 7:
P A Note to Students, Teachers, and
Page 10 and 11:
Contents 5 Graphical Exploraons of
Page 12 and 13:
Chapter 1 Systems of Linear Equaons
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Chapter 2 Matrix Arithmec In the pa
Page 58 and 59:
Chapter 2 Matrix Arithmec ⎡ 3 6
Page 60 and 61:
Chapter 2 Matrix Arithmec by the nu
Page 62 and 63:
Chapter 2 Matrix Arithmec means? Yo
Page 64 and 65:
Chapter 2 . Matrix Arithmec ⎡ ⎤
Page 66 and 67:
Chapter 2 Matrix Arithmec using the
Page 68 and 69:
Chapter 2 Matrix Arithmec the numbe
Page 70 and 71:
Chapter 2 Matrix Arithmec BB = BC,
Page 72 and 73:
Chapter 2 Matrix Arithmec identy ma
Page 74 and 75:
Chapter 2 Matrix Arithmec (a) Give
Page 76 and 77:
Chapter 2 Matrix Arithmec 2.3 Visua
Page 78 and 79:
Chapter 2 Matrix Arithmec Vector Ad
Page 80 and 81:
Chapter 2 Matrix Arithmec Scalar Mu
Page 82 and 83:
Chapter 2 Matrix Arithmec S To draw
Page 84 and 85:
Chapter 2 Matrix Arithmec . Ḋefin
Page 86 and 87:
Chapter 2 Matrix Arithmec y A⃗x A
Page 88 and 89:
Chapter 2 Matrix Arithmec Of course
Page 90 and 91:
Chapter 2 Matrix Arithmec 2.4 Vecto
Page 92 and 93:
Chapter 2 Matrix Arithmec In previo
Page 94 and 95:
Chapter 2 Matrix Arithmec y ⃗v .
Page 96 and 97:
Chapter 2 Matrix Arithmec Again, In
Page 98 and 99:
Chapter 2 Matrix Arithmec equaon is
Page 100 and 101:
Chapter 2 Matrix Arithmec . Key Ide
Page 102 and 103:
Chapter 2 Matrix Arithmec “x 3
Page 104 and 105:
Chapter 2 Matrix Arithmec . Example
Page 106 and 107:
Chapter 2 Matrix Arithmec [ ] 1 −
Page 108 and 109:
Chapter 2 Matrix Arithmec We know h
Page 110 and 111:
Chapter 2 Matrix Arithmec Noce also
Page 112 and 113:
Chapter 2 Matrix Arithmec from abov
Page 114 and 115:
Chapter 2 Matrix Arithmec 4. T/F: I
Page 116 and 117:
Chapter 2 Matrix Arithmec Since mat
Page 118 and 119:
Chapter 2 Matrix Arithmec We have a
Page 120 and 121:
Chapter 2 Matrix Arithmec . Ṫheor
Page 122 and 123:
Chapter 2 Matrix Arithmec ⎡ −15
Page 124 and 125:
Chapter 2 Matrix Arithmec is that t
Page 126 and 127:
Chapter 2 Matrix Arithmec We now ap
Page 128 and 129:
Chapter 2 Matrix Arithmec more to b
Page 130 and 131:
Chapter 2 Matrix Arithmec can cause
Page 132 and 133:
Chapter 3 Operaons on Matrices .
Page 134 and 135:
Chapter 3 Operaons on Matrices Ther
Page 136 and 137:
Chapter 3 Operaons on Matrices Find
Page 138 and 139:
Chapter 3 Operaons on Matrices Let
Page 140 and 141:
Chapter 3 Operaons on Matrices oper
Page 142 and 143:
Chapter 3 Operaons on Matrices The
Page 144 and 145:
Chapter 3 Operaons on Matrices We e
Page 146 and 147:
Chapter 3 Operaons on Matrices In t
Page 148 and 149:
Chapter 3 Operaons on Matrices S To
Page 150 and 151:
Chapter 3 Operaons on Matrices We
Page 152 and 153:
Chapter 3 Operaons on Matrices C 1,
Page 154 and 155:
Chapter 3 Operaons on Matrices C 1,
Page 156 and 157: Chapter 3 Operaons on Matrices 21.
Page 158 and 159: Chapter 3 Operaons on Matrices S At
Page 160 and 161: Chapter 3 Operaons on Matrices .
Page 162 and 163: Chapter 3 Operaons on Matrices 1 2
Page 164 and 165: Chapter 3 Operaons on Matrices Obvi
Page 166 and 167: Chapter 3 Operaons on Matrices . Ke
Page 168 and 169: Chapter 3 Operaons on Matrices ⎡
Page 170 and 171: Chapter 3 Operaons on Matrices S Ru
Page 172 and 173: Chapter 3 Operaons on Matrices read
Page 174 and 175: Chapter 4 Eigenvalues and Eigenvect
Page 186 and 187: Chapter 4 . Eigenvalues and Eigenve
Page 197 and 198: 5 . G E V We already looked at the
Page 199 and 200: 5.1 Transformaons of the Cartesian
Page 205: 5.1 Transformaons of the Cartesian
Page 213 and 214: 5.2 Properes of Linear Transformaon
Page 225 and 226: 5.3 Visualizing Vectors: Vectors in
Page 237 and 238: A . S T S P Chapter 1 Secon 1.1 1.
Page 239 and 240: [ ] 9 −7 5. 11 −6 [ ] −14 7.
Page 241 and 242: 1. Mulply A⃗u and A⃗v to verify
Page 243 and 244: 21. A is diagonal, as is A T . ⎡
Page 245 and 246: (d) -1 (e) 0 ⎡ 5. (a) λ 1 = −4
Page 247 and 248: Index ansymmetric, 128 augmented ma
show all

Fundamentals of Matrix Algebra, 2011a

Create successful ePaper yourself

Delete template?

Save as template?