Linear Algebra for Earth Scientists

1. Rows and Columns. 1.1. The Three-Point Problem. 1.2. Can we solve this more easily using Linear Algebra? 1.3. Thinking about Columns and Vectors. 1.4. Vectors. 1.5. A Geochemical Example. 1.6. Summary. 1.7. Exercises. 2. Matrix multiplication. 2.1. Multiplying the Linear Algebra Way. 2.2. Transpose of a Vector or Matrix. 2.3. Linear Combinations. 2.4. Linear Transformations. 2.5. Geologic Transformations. 2.6. What more can we do with Matrices? 2.7. More about 3D Rotations. 2.8. Summary. 2.9. Exercises. 3. Solving Ax = b. 3.1. Elimination. 3.2. Elimination Matrices and Elementary Row Operations. 3.3. Elementary Matrices and their Inverses. 3.4. Our First Factorizations — A = LU = LDU. 3.5. Gauss-Jordan Elimination and Computing A-1. 3.6. The Three-Point Problem Many Ways. 3.7. Reworking the Geochemical Example. 3.8. Summary. 3.9. Exercises. 4 When does Ax = b have a solution? 4.1. Slip with Different Number of Faults. 4.2. The Three-Point Problem with more Points. 4.3. The Vector b must be in the Column space of A to find x. 4.4. Exploring the vector spaces and Subspaces of Ax=b. 4.5. Rank and Size of a determines if we can solve Ax = b. 4.6. Summary. 4.7. Exercises. 5. Solving Ax = b when there is no solution. 5.1. Projection. 5.2. Solving Projection for Lines and Planes. 5.3. Can we fit something other than a Line or Plane. 5.4. Summary. 5.5. Exercises. 6. Determinants and Orthogonality. 6.1. Determinants, Strain, and Geologic Transformations. 6.2. Strain, Determinants, and Basic Matrix Operations. 6.3. Cross Products. 6.4. Orthogonality and Gram-Schmidt Process. 6.5. Decomposition - Our next Factorization. 6.6. Summary. 6.7. Exercise. 7. An Earth Scientist's view of the characteristics of Eigenvalues and Eigenvectors. 7.1. Motivation for Eigenanalysis. 7.2. Eigenvalues and Eigenvectors. 7.3. Looking at Data and not Ax=b. 7.4. Diagonalization into X?x-1. 7.5. A. Detailed look at what XT, ?, and X Do. 7.6. Summary. 7.7. Exercises. 8. Change of Basis, Eigenbasis, and Quadratic Forms. 8.1. Change of Basis. 8.2. What do these results mean for the Eigenbasis? 8.3. The Geologist's Dilemma - Setting a Bruntonr Compass. 8.4. A Nice Gneiss. 8.5. Stressing the use of Eigenanalysis. 8.6. Quadratic Forms. 8.7. Summary. 8.8. Exercises. 9. Singular Value Decomposition. 9.1. What is Singular Value Decomposition or SVD? 9.2. SVD for any Matrix. 9.3. When do Simple and Pure Shear produce the same strain Ellipse? 9.4. Using SVD to solve Ax = b. 9.5. Condition Numbers, or, when good Matrices go bad. 9.6. Summary. 9.7. Exercises.