Contents
Preface
1 Linear Algebra
1 Vectors in Rn
2 Dot product and angle between vectors in Rn
3 Subspaces and linear dependence of vectors
4 Gaussian Elimination and the Linear Dependence Lemma
5 The Basis Theorem
6 Matrices
7 Rank and the Rank-Nullity Theorem
8 Orthogonal complements and orthogonal projection
9 Row Echelon Form of a Matrix
10 Inhomogeneous systems
2 Analysis in Rn
1 Open and closed sets in Euclidean Space
2 Bolzano-Weierstrass, Limits and Continuity in Rn
3 Differentiability
4 Directional Derivatives, Partial Derivatives, and Gradient
5 Chain Rule
6 Higher-order partial derivatives
7 Second derivative test for extrema of multivariable function
8 Curves in Rn
9 Submanifolds of Rn and tangential gradients
3 More I,inear Algebra
1 Permutations
2 Determinants
3 Inverse of a Square Matrix
4 Computing the Inverse
5 Orthonormal Basis and Gram-Schmidt
6 Matrix Representations of Linear Transformations
7 Eigenvalues and the Spectral Theorem
4 More Analysis in Rn
1 Contraction Mapping Principle
2 Inverse Function Theorem
3 Implicit Function Theorem
A Introductory Lectures on Real Analysis
Lecture 1: The Real Numbers
Lecture 2: Sequences of Real Numbers and the Bolzano-Weierstrass Theorem
Lecture 3: Continuous Functions
Lecture 4: Series of Real Numbers
Lecture 5: Power Series
Lecture 6: Taylor Series Representations
Lecture 7: Complex Series, Products of Series, and Complex Exponential Series
Lecture 8: Fourier Series
Lecture 9: Pointwise Convergence of Trigonometric Fourier Series
Index
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