### Coding the Matrix: Linear Algebra through Applications to Computer Science

An attractive advent to vectors and matrices and the algorithms that function on them, meant for the scholar who is aware the best way to application. Mathematical recommendations and computational difficulties are stimulated through purposes in laptop technological know-how. The reader learns via doing, writing courses to enforce the mathematical thoughts and utilizing them to hold out initiatives and discover the functions. Examples comprise: error-correcting codes, variations in snap shots, face detection, encryption and secret-sharing, integer factoring, removal point of view from a picture, PageRank (Google's rating algorithm), and melanoma detection from telephone positive aspects. A significant other site,

`codingthematrix.com`

offers information and aid code. lots of the assignments should be auto-graded on-line. Over 200 illustrations, together with a variety of suitable xkcd comics.

Chapters: The Function, The Field, The Vector, The Vector Space, The Matrix, The Basis, Dimension, Gaussian Elimination, The internal Product, Special Bases, The Singular price Decomposition, The Eigenvector, The Linear Program

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Four. eleven. five Column vector and row vector . . . . . . . . . . . . . . . . . . . . . . . . four. eleven. 6 each vector is interpreted as a column vector . . . . . . . . . . . . . . . four. eleven. 7 Linear mixtures of linear mixtures revisited . . . . . . . . . . . . four. 2 four. three four. four four. five iv 191 192 193 193 194 194 195 197 198 2 hundred two hundred 201 204 206 207 208 208 210 211 212 212 213 214 214 215 215 218 218 219 221 221 222 223 224 225 225 226 229 233 236 237 238 238 v CONTENTS four. 12 internal product and outer product . . . . . . . . . . . . . . . . . . four. 12. 1 internal product . . . . . . . . . . . . . .

Four. eleven. 2 Graphs, prevalence matrices, and counting paths . . . . . . . . . . . . . . . four. eleven. three Matrix-matrix multiplication and serve as composition . . . . . . . . . . four. eleven. four Transpose of matrix-matrix product . . . . . . . . . . . . . . . . . . . . . four. eleven. five Column vector and row vector . . . . . . . . . . . . . . . . . . . . . . . . four. eleven. 6 each vector is interpreted as a column vector . . . . . . . . . . . . . . . four. eleven. 7 Linear combos of linear mixtures revisited . . . . . . . . . . . . four. 2 four. three four. four four. five iv 191 192 193 193 194 194 195 197 198 2 hundred 2 hundred 201 204 206 207 208 208 210 211 212 212 213 214 214 215 215 218 218 219 221 221 222 223 224 225 225 226 229 233 236 237 238 238 v CONTENTS four.

Un + vn ] · [w1 , . . . , wn ] (u1 + v1 )w1 + · · · + (un + vn )wn u1 w 1 + v 1 w 1 + · · · + un w n + v n w n (u1 w1 + · · · + un wn ) + (v1 w1 + · · · + vn wn ) [u1 , . . . , un ] · [w1 , . . . , wn ] + [v1 , . . . , vn ] · [w1 , . . . , wn ] challenge 2. nine. 26: express by way of giving a counterexample that (u + v) · (w + x) = u · w + v · x isn't really precise. instance 2. nine. 27: We first provide an instance of the distributive estate for vectors over the 125 bankruptcy 2. THE VECTOR reals: [27, 37, forty seven] · [2, 1, 1] = [20, 30, forty] · [2, 1, 1] + [7, 7, 7] · [2, 1, 1]: • 20 2 20 · 2 + 30 1 30 · 1 + 7 2 7·2 7 1 7·1 • • 2.

Five. 7. 1 specialty of illustration by way of a foundation . . . . . . . . . . . . . . . five. eight swap of foundation, first glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. eight. 1 The functionality from illustration to vector . . . . . . . . . . . . . . . . . five. eight. 2 From one illustration to a different . . . . . . . . . . . . . . . . . . . . . five. nine standpoint rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. nine. 1 issues on this planet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . five. nine. 2 The digital camera and the picture aircraft . . . . . . . . . . . . . . . . . . . . . . . five. nine. three The digital camera coordinate approach .

During this bankruptcy, we'll come upon the concept that of vector areas, an idea that underlies the solutions and every little thing else we do during this booklet. three. 1 three. 1. 1 Linear mix Definition of linear mix Definition three. 1. 1: feel v1 , . . . , vn are vectors. We outline a linear blend of v1 , . . . , vn to be a sum α1 v1 + · · · + αn vn the place α1 , . . . , αn are scalars. during this context, we check with α1 , . . . , αn because the coeﬃcients during this linear mix. particularly, α1 is the coeﬃcient of v1 within the linear blend, α2 is the coeﬃcient of v2 , and so forth.