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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