Top
| Linear Algebra | p. 1 |
| Vectors in R" | p. 1 |
| Dot product and angle between vectors in R" | p. 3 |
| Subspaces and linear dependence of vectors | p. 5 |
| Gaussian Elimination and the Linear Dependence Lemma | p. 7 |
| The Basis Theorem | p. 11 |
| Matrices | p. 13 |
| Rank and the Rank-Nullity Theorem | p. 15 |
| Orthogonal complements and orthogonal projection | p. 18 |
| Row Echelon Form of a Matrix | p. 22 |
| Inhomogeneous systems | p. 27 |
| Analysis in R" | p. 31 |
| Open and closed sets in Euclidean Space | p. 31 |
| Bolzano-Weierstrass, Limits and Continuity in R" | p. 33 |
| Differentiability | p. 35 |
| Directional Derivatives, Partial Derivatives, and Gradient | p. 37 |
| Chain Rule | p. 41 |
| Higher-order partial derivatives | p. 42 |
| Second derivative test for extrema of multivariable function | p. 44 |
| Curves in R" | p. 48 |
| Submanifolds of R" and tangential gradients | p. 53 |
| More Linear Algebra | p. 61 |
| Permutations | p. 61 |
| Determinants | p. 64 |
| Inverse of a Square Matrix | p. 69 |
| Computing the Inverse | p. 72 |
| Orthonormal Basis and Gram-Schmidt | p. 73 |
| Matrix Representations of Linear Transformations | p. 75 |
| Eigenvalues and the Spectral Theorem | p. 76 |
| More Analysis in R" | p. 81 |
| Contraction Mapping Principle | p. 81 |
| Inverse Function Theorem | p. 82 |
| Implicit Function Theorem | p. 84 |
| Introductory Lectures on Real Analysis | p. 87 |
| The Real Numbers | p. 87 |
| Sequences of Real Numbers and the Bolzano-Weierstrass Theorem | p. 91 |
| Continuous Functions | p. 96 |
| Series of Real Numbers | p. 100 |
| Power Series | p. 105 |
| Taylor Series Representations | p. 108 |
| Complex Series, Products of Series, and Complex Exponential Series | p. 113 |
| Fourier Series | p. 116 |
| Pointwise Convergence of Trigonometric Fourier Series | p. 121 |
| Index | p. 127 |
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