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Linear Algebra, Theory And Applications - Brigham Young …

Linear algebra , Theory And Applications Kenneth Kuttler September 12, 2017. 2. Contents 1 Preliminaries 11. Sets And Set Notation .. 11. Functions .. 12. The Number Line And algebra Of The Real Numbers .. 12. Ordered fields .. 14. The Complex Numbers .. 15. The Fundamental Theorem Of algebra .. 18. Exercises .. 20. Completeness of R .. 21. Well Ordering And Archimedean Property .. 22. Division .. 24. Systems Of Equations .. 28. Exercises .. 33. Fn .. 34. algebra in Fn .. 34. Exercises .. 35. The Inner Product In Fn .. 35. What Is Linear algebra ? .. 38. Exercises.

customary to write 3 2f1;2;3;8g:9 2f= 1;2;3;8gmeans 9 is not an element of f1;2;3;8g: Sometimes a rule speci es a set. For example you could specify a set as all integers larger

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Transcription of Linear Algebra, Theory And Applications - Brigham Young …

1 Linear algebra , Theory And Applications Kenneth Kuttler September 12, 2017. 2. Contents 1 Preliminaries 11. Sets And Set Notation .. 11. Functions .. 12. The Number Line And algebra Of The Real Numbers .. 12. Ordered fields .. 14. The Complex Numbers .. 15. The Fundamental Theorem Of algebra .. 18. Exercises .. 20. Completeness of R .. 21. Well Ordering And Archimedean Property .. 22. Division .. 24. Systems Of Equations .. 28. Exercises .. 33. Fn .. 34. algebra in Fn .. 34. Exercises .. 35. The Inner Product In Fn .. 35. What Is Linear algebra ? .. 38. Exercises.

2 38. 2 Linear Transformations 39. Matrices .. 39. The ij th Entry Of A Product .. 44. Digraphs .. 46. Properties Of Matrix Multiplication .. 48. Finding The Inverse Of A Matrix .. 51. Exercises .. 54. Linear Transformations .. 56. Some Geometrically Defined Linear Transformations .. 58. The Null Space Of A Linear Transformation .. 61. Subspaces And Spans .. 62. An Application To Matrices .. 67. Matrices And Calculus .. 68. The Coriolis Acceleration .. 69. The Coriolis Acceleration On The Rotating Earth .. 73. Exercises .. 78. 3. 4 CONTENTS. 3 Determinants 85.

3 Basic Techniques And Properties .. 85. Exercises .. 89. The Mathematical Theory Of Determinants .. 91. The Function sgn .. 91. The Definition Of The Determinant .. 93. A Symmetric Definition .. 95. Basic Properties Of The Determinant .. 96. Expansion Using Cofactors .. 97. A Formula For The Inverse .. 99. Rank Of A Matrix .. 101. Summary Of Determinants .. 103. The Cayley Hamilton Theorem .. 104. Block Multiplication Of Matrices .. 106. Exercises .. 109. 4 Row Operations 113. Elementary Matrices .. 113. The Rank Of A Matrix .. 119. The Row Reduced Echelon Form.

4 121. Rank And Existence Of Solutions To Linear Systems .. 124. Fredholm Alternative .. 125. Exercises .. 126. 5 Some Factorizations 131. LU Factorization .. 131. Finding An LU Factorization .. 131. Solving Linear Systems Using An LU Factorization .. 133. The P LU Factorization .. 134. Justification For The Multiplier Method .. 136. Existence For The P LU Factorization .. 137. The QR Factorization .. 138. Exercises .. 141. 6 Spectral Theory 145. Eigenvalues And Eigenvectors Of A Matrix .. 145. Some Applications Of Eigenvalues And Eigenvectors .. 153. Exercises.

5 156. Schur's Theorem .. 162. Trace And Determinant .. 169. Quadratic Forms .. 170. Second Derivative Test .. 171. The Estimation Of Eigenvalues .. 175. Advanced Theorems .. 176. Exercises .. 179. 7 Vector Spaces And Fields 189. Vector Space Axioms .. 189. Subspaces And Bases .. 190. Basic Definitions .. 190. A Fundamental Theorem .. 190. The Basis Of A Subspace .. 195. CONTENTS 5. Lots Of Fields .. 195. Irreducible Polynomials .. 195. Polynomials And Fields .. 199. The Algebraic Numbers .. 205. The Lindemannn Weierstrass Theorem And Vector Spaces .. 208. Exercises.

6 209. 8 Linear Transformations 215. Matrix Multiplication As A Linear Transformation .. 215. L (V, W ) As A Vector Space .. 215. The Matrix Of A Linear Transformation .. 217. Rotations About A Given Vector .. 224. The Euler Angles .. 226. Eigenvalues And Eigenvectors Of Linear Transformations .. 228. Exercises .. 229. 9 Canonical Forms 233. A Theorem Of Sylvester, Direct Sums .. 233. Direct Sums, Block Diagonal Matrices .. 236. Cyclic Sets .. 239. Nilpotent Transformations .. 243. The Jordan Canonical Form .. 246. Exercises .. 249. The Rational Canonical Form.

7 254. Uniqueness .. 256. Exercises .. 261. 10 Markov Processes 263. Regular Markov Matrices .. 263. Migration Matrices .. 267. Absorbing States .. 267. Exercises .. 270. 11 Inner Product Spaces 273. General Theory .. 273. The Gram Schmidt Process .. 275. Riesz Representation Theorem .. 278. The Tensor Product Of Two Vectors .. 281. Least Squares .. 283. Fredholm Alternative Again .. 284. Exercises .. 285. The Determinant And Volume .. 289. Exercises .. 291. 12 Self Adjoint Operators 293. Simultaneous Diagonalization .. 293. Schur's Theorem .. 296. Spectral Theory Of Self Adjoint Operators.

8 298. Positive And Negative Linear Transformations .. 302. The Square Root .. 304. Fractional Powers .. 305. Square Roots And Polar Decompositions .. 306. 6 CONTENTS. An Application To Statistics .. 309. The Singular Value Decomposition .. 311. In The Frobenius Norm .. 313. Squares And Singular Value Decomposition .. 315. Moore Penrose Inverse .. 315.. 319. 13 Norms 323. The p Norms .. 329. The Condition Number .. 331. The Spectral Radius .. 333. Series And Sequences Of Linear Operators .. 335. Iterative Methods For Linear Systems .. 340. Theory Of Convergence.

9 345. Exercises .. 348. 14 Numerical Methods, Eigenvalues 355. The Power Method For Eigenvalues .. 355. The Shifted Inverse Power Method .. 358. The Explicit Description Of The Method .. 359. Complex Eigenvalues .. 364. Rayleigh Quotients And Estimates for Eigenvalues .. 365. The QR Algorithm .. 369. Basic Properties And Definition .. 369. The Case Of Real Eigenvalues .. 372. The QR Algorithm In The General Case .. 376. Exercises .. 383. A Matrix Calculator On The Web 385. Use Of Matrix Calculator On Web .. 385. B Positive Matrices 387. C Functions Of Matrices 395.

10 D Di erential Equations 401. Theory Of Ordinary Di erential Equations .. 401. Linear Systems .. 402. Local Solutions .. 403. First Order Linear Systems .. 405. Geometric Theory Of Autonomous Systems .. 413. General Geometric Theory .. 416. The Stable Manifold .. 418. E Compactness And Completeness 423. The Nested Interval Lemma .. 423. Convergent Sequences, Sequential Compactness .. 424. CONTENTS 7. F Some Topics Flavored With Linear algebra 427. The Symmetric Polynomial Theorem .. 427. The Fundamental Theorem Of algebra .. 429. Transcendental Numbers .. 433.


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