**Complete linear algebra: theory and implementation in code**

English | MP4 | AVC 1280×720 | AAC 44KHz 2ch | 33 Hours | 9.69 GB

Learn concepts in linear algebra and matrix analysis, and implement them in MATLAB and Python.

You need to learn linear algebra!

Linear algebra is perhaps the most important branch of mathematics for computational sciences, including machine learning, AI, data science, statistics, simulations, computer graphics, multivariate analyses, matrix decompositions, signal processing, and so on.

You need to know applied linear algebra, not just abstract linear algebra!

The way linear algebra is presented in 30-year-old textbooks is different from how professionals use linear algebra in computers to solve real-world applications in machine learning, data science, statistics, and signal processing. For example, the “determinant” of a matrix is important for linear algebra theory, but should you actually use the determinant in practical applications? The answer may surprise you, and it’s in this course!

If you are interested in learning the mathematical concepts linear algebra and matrix analysis, but also want to apply those concepts to data analyses on computers (e.g., statistics or signal processing), then this course is for you! You’ll see all the maths concepts implemented in MATLAB and in Python.

Unique aspects of this course

- Clear and comprehensible explanations of concepts and theories in linear algebra.
- Several distinct explanations of the same ideas, which is a proven technique for learning.
- Visualization using graphs, numbers, and spaces that strengthens the geometric intuition of linear algebra.
- Implementations in MATLAB and Python. Com’on, in the real world, you never solve math problems by hand! You need to know how to implement math in software!
- Beginning to intermediate topics, including vectors, matrix multiplications, least-squares projections, eigendecomposition, and singular-value decomposition.
- Strong focus on modern applications-oriented aspects of linear algebra and matrix analysis.
- Intuitive visual explanations of diagonalization, eigenvalues and eigenvectors, and singular value decomposition.
- Improve your coding skills! You do need to have a little bit of coding experience for this course (I do not teach elementary Python or MATLAB), but you will definitely improve your scientific and data analysis programming skills in this course. Everything is explained in MATLAB and in Python (mostly using numpy and matplotlib; also sympy and scipy and some other relevant toolboxes).

Benefits of learning linear algebra

- Understand statistics including least-squares, regression, and multivariate analyses.
- Improve mathematical simulations in engineering, computational biology, finance, and physics.
- Understand data compression and dimension-reduction (PCA, SVD, eigendecomposition).
- Understand the math underlying machine learning and linear classification algorithms.
- Deeper knowledge of signal processing methods, particularly filtering and multivariate subspace methods.
- Explore the link between linear algebra, matrices, and geometry.
- Gain more experience implementing math and understanding machine-learning concepts in Python and MATLAB.

Why I am qualified to teach this course:

I have been using linear algebra extensively in my research and teaching (in MATLAB and Python) for many years. I have written several textbooks about data analysis, programming, and statistics, that rely extensively on concepts in linear algebra.

What you’ll learn

- Understand theoretical concepts in linear algebra, including proofs
- Implement linear algebra concepts in scientific programming languages (MATLAB, Python)
- Apply linear algebra concepts to real datasets
- Ace your linear algebra exam!
- Apply linear algebra on computers with confidence
- Gain additional insights into solving problems in linear algebra, including homeworks and applications
- Be confident in learning advanced linear algebra topics
- Understand some of the important maths underlying machine learning
- Manually corrected closed-captions

**Table of Contents**

**Introductions**

1 What is linear algebra

2 Linear algebra applications

3 An enticing start to a linear algebra course!

4 How best to learn from this course

5 Maximizing your Udemy experience

6 Using MATLAB, Octave, or Python in this course

**Vectors**

7 Exercises + code

8 Algebraic and geometric interpretations of vectors

9 Vector addition and subtraction

10 Vector-scalar multiplication

11 Vector-vector multiplication the dot product

12 Dot product properties associative, distributive, commutative

13 Code challenge dot products with matrix columns

14 Vector length

15 Dot product geometry sign and orthogonality

16 Code challenge dot product sign and scalar multiplication

17 Code challenge is the dot product commutative

18 Vector Hadamard multiplication

19 Outer product

20 Vector cross product

21 Vectors with complex numbers

22 Hermitian transpose (a.k.a. conjugate transpose)

23 Interpreting and creating unit vectors

24 Code challenge dot products with unit vectors

25 Dimensions and fields in linear algebra

26 Subspaces

27 Subspaces vs. subsets

28 Span

29 Linear independence

30 Basis

**Introduction to matrices**

31 Exercises + code

32 Matrix terminology and dimensionality

33 A zoo of matrices

34 Matrix addition and subtraction

35 Matrix-scalar multiplication

36 Code challenge is matrix-scalar multiplication a linear operation

37 Transpose

38 Complex matrices

39 Diagonal and trace

40 Code challenge linearity of trace

41 Broadcasting matrix arithmetic

**Matrix multiplications**

42 Exercises + code

43 Introduction to standard matrix multiplication

44 Four ways to think about matrix multiplication

45 Code challenge matrix multiplication by layering

46 Matrix multiplication with a diagonal matrix

47 Order-of-operations on matrices

48 Matrix-vector multiplication

49 D transformation matrices

50 Code challenge Pure and impure rotation matrices

51 Code challenge Geometric transformations via matrix multiplications

52 Additive and multiplicative matrix identities

53 Additive and multiplicative symmetric matrices

54 Hadamard (element-wise) multiplication

55 Code challenge symmetry of combined symmetric matrices

56 Multiplication of two symmetric matrices

57 Code challenge standard and Hadamard multiplication for diagonal matrices

58 Code challenge Fourier transform via matrix multiplication!

59 Frobenius dot product

60 Matrix norms

61 Code challenge conditions for self-adjoint

62 What about matrix division

**Matrix rank**

63 Exercises + code

64 Rank concepts, terms, and applications

65 Computing rank theory and practice

66 Rank of added and multiplied matrices

67 Code challenge reduced-rank matrix via multiplication

68 Code challenge scalar multiplication and rank

69 Rank of A^TA and AA^T

70 Code challenge rank of multiplied and summed matrices

71 Making a matrix full-rank by shifting

72 Code challenge is this vector in the span of this set

73 Course tangent self-accountability in online learning

**Matrix spaces**

74 Exercises + code

75 Column space of a matrix

76 Column space, visualized in code

77 Row space of a matrix

78 Null space and left null space of a matrix

79 Columnleft-null and rownull spaces are orthogonal

80 Dimensions of columnrownull spaces

81 Example of the four subspaces

82 More on Ax=b and Ax=0

**Solving systems of equations**

83 Exercises + code

84 Systems of equations algebra and geometry

85 Converting systems of equations to matrix equations

86 Gaussian elimination

87 Echelon form and pivots

88 Reduced row echelon form

89 Code challenge RREF of matrices with different sizes and ranks

90 Matrix spaces after row reduction

**Matrix determinant**

91 Exercises

92 Determinant concept and applications

93 Determinant of a 2×2 matrix

94 Code challenge determinant of small and large singular matrices

95 Determinant of a 3×3 matrix

96 Code challenge large matrices with row exchanges

97 Find matrix values for a given determinant

98 Code challenge determinant of shifted matrices

99 Code challenge determinant of matrix product

**Matrix inverse**

100 Exercises + code

101 Matrix inverse Concept and applications

102 Computing the inverse in code

103 Inverse of a 2×2 matrix

104 The MCA algorithm to compute the inverse

105 Code challenge Implement the MCA algorithm!!

106 Computing the inverse via row reduction

107 Code challenge inverse of a diagonal matrix

108 Left inverse and right inverse

109 One-sided inverses in code

110 Proof the inverse is unique

111 Pseudo-inverse, part 1

112 Code challenge pseudoinverse of invertible matrices

**Projections and orthogonalization**

113 Exercises + code

114 Projections in R^2

115 Projections in R^N

116 Orthogonal and parallel vector components

117 Code challenge decompose vector to orthogonal components

118 Orthogonal matrices

119 Gram-Schmidt procedure

120 QR decomposition

121 Code challenge Gram-Schmidt algorithm

122 Matrix inverse via QR decomposition

123 Code challenge Inverse via QR

124 Code challenge Prove and demonstrate the Sherman-Morrison inverse

125 Code challenge A^TA = R^TR

**Least-squares for model-fitting in statistics**

126 Exercises + code

127 Introduction to least-squares

128 Least-squares via left inverse

129 Least-squares via orthogonal projection

130 Least-squares via row-reduction

131 Model-predicted values and residuals

132 Least-squares application 1

133 Least-squares application 2

134 Code challenge Least-squares via QR decomposition

**Eigendecomposition**

135 Exercises + code

136 What are eigenvalues and eigenvectors

137 Finding eigenvalues

138 Shortcut for eigenvalues of a 2×2 matrix

139 Code challenge eigenvalues of diagonal and triangular matrices

140 Code challenge eigenvalues of random matrices

141 Finding eigenvectors

142 Eigendecomposition by hand two examples

143 Diagonalization

144 Matrix powers via diagonalization

145 Code challenge eigendecomposition of matrix differences

146 Eigenvectors of distinct eigenvalues

147 Eigenvectors of repeated eigenvalues

148 Eigendecomposition of symmetric matrices

149 Eigenlayers of a matrix

150 Code challenge reconstruct a matrix from eigenlayers

151 Eigendecomposition of singular matrices

152 Code challenge trace and determinant, eigenvalues sum and product

153 Generalized eigendecomposition

154 Code challenge GED in small and large matrices

**Singular value decomposition**

155 Exercises + code

156 Singular value decomposition (SVD)

157 Code challenge SVD vs. eigendecomposition for square symmetric matrices

158 Relation between singular values and eigenvalues

159 Code challenge U from eigendecomposition of A^TA

160 Code challenge A^TA, Av, and singular vectors

161 SVD and the four subspaces

162 Spectral theory of matrices

163 SVD for low-rank approximations

164 Convert singular values to percent variance

165 Code challenge When is UV^T valid, what is its norm, and is it orthogonal

166 SVD, matrix inverse, and pseudoinverse

167 Condition number of a matrix

168 Code challenge Create matrix with desired condition number

**Quadratic form and definiteness**

169 Exercises + code

170 The quadratic form in algebra

171 The quadratic form in geometry

172 The normalized quadratic form

173 Code challenge Visualize the normalized quadratic form

174 Eigenvectors and the quadratic form surface

175 Application of the normalized quadratic form PCA

176 Quadratic form of generalized eigendecomposition

177 Matrix definiteness, geometry, and eigenvalues

178 Proof A^TA is always positive (semi)definite

179 Proof Eigenvalues and matrix definiteness

**Bonus section**

180 Bonus lecture

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