Complete linear algebra: theory and implementation

32h 53m 26s
English
Paid
April 25, 2024

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!

More

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! 

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.

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.

Why I am qualified to teach this course:

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

Requirements:
  • Basic understanding of high-school algebra (e.g., solve for x in 2x=5)
  • Interest in learning about matrices and vectors!
  • (optional) Computer with MATLAB, Octave, or Python (or Jupyter)
Who this course is for:
  • Anyone interested in learning about matrices and vectors
  • Students who want supplemental instruction/practice for a linear algebra course
  • Engineers who want to refresh their knowledge of matrices and decompositions
  • Biologists who want to learn more about the math behind computational biology
  • Data scientists (linear algebra is everywhere in data science!)
  • Statisticians
  • Someone who wants to know the important math underlying machine learning
  • Someone who studied theoretical linear algebra and who wants to implement concepts in computers
  • Computational scientists (statistics, biological, engineering, neuroscience, psychology, physics, etc.)
  • Someone who wants to learn about eigendecomposition, diagonalization, and singular value decomposition!

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 *

Watch Online Complete linear algebra: theory and implementation

Join premium to watch
Go to premium
# Title Duration
1 What is linear algebra? 08:04
2 Linear algebra applications 05:58
3 An enticing start to a linear algebra course! 13:14
4 How best to learn from this course 04:00
5 Maximizing your Udemy experience 07:58
6 Using MATLAB, Octave, or Python in this course 07:31
7 Algebraic and geometric interpretations of vectors 12:46
8 Vector addition and subtraction 08:27
9 Vector-scalar multiplication 09:08
10 Vector-vector multiplication: the dot product 10:12
11 Dot product properties: associative, distributive, commutative 18:56
12 Code challenge: dot products with matrix columns 08:46
13 Vector length 06:43
14 Dot product geometry: sign and orthogonality 23:39
15 Code challenge: dot product sign and scalar multiplication 12:06
16 Code challenge: is the dot product commutative? 09:33
17 Vector Hadamard multiplication 03:44
18 Outer product 10:18
19 Vector cross product 09:06
20 Vectors with complex numbers 08:18
21 Hermitian transpose (a.k.a. conjugate transpose) 16:22
22 Interpreting and creating unit vectors 07:59
23 Code challenge: dot products with unit vectors 13:34
24 Dimensions and fields in linear algebra 07:55
25 Subspaces 15:51
26 Subspaces vs. subsets 05:48
27 Span 13:30
28 Linear independence 15:35
29 Basis 11:52
30 Matrix terminology and dimensionality 08:15
31 A zoo of matrices 17:20
32 Matrix addition and subtraction 08:29
33 Matrix-scalar multiplication 02:34
34 Code challenge: is matrix-scalar multiplication a linear operation? 07:29
35 Transpose 10:25
36 Complex matrices 01:52
37 Diagonal and trace 09:08
38 Code challenge: linearity of trace 09:38
39 Broadcasting matrix arithmetic 14:14
40 Introduction to standard matrix multiplication 10:28
41 Four ways to think about matrix multiplication 11:56
42 Code challenge: matrix multiplication by layering 09:46
43 Matrix multiplication with a diagonal matrix 03:43
44 Order-of-operations on matrices 08:16
45 Matrix-vector multiplication 16:44
46 2D transformation matrices 15:33
47 Code challenge: Pure and impure rotation matrices 12:39
48 Code challenge: Geometric transformations via matrix multiplications 15:59
49 Additive and multiplicative matrix identities 06:20
50 Additive and multiplicative symmetric matrices 15:17
51 Hadamard (element-wise) multiplication 05:01
52 Code challenge: symmetry of combined symmetric matrices 12:04
53 Multiplication of two symmetric matrices 13:22
54 Code challenge: standard and Hadamard multiplication for diagonal matrices 06:28
55 Code challenge: Fourier transform via matrix multiplication! 11:21
56 Frobenius dot product 11:17
57 Matrix norms 18:12
58 Code challenge: conditions for self-adjoint 11:53
59 What about matrix division? 04:25
60 Rank: concepts, terms, and applications 10:51
61 Computing rank: theory and practice 23:02
62 Rank of added and multiplied matrices 11:47
63 Code challenge: reduced-rank matrix via multiplication 10:39
64 Code challenge: scalar multiplication and rank 12:11
65 Rank of A^TA and AA^T 10:42
66 Code challenge: rank of multiplied and summed matrices 07:07
67 Making a matrix full-rank by "shifting" 14:13
68 Code challenge: is this vector in the span of this set? 11:47
69 Course tangent: self-accountability in online learning 03:04
70 Column space of a matrix 13:30
71 Column space, visualized in code 06:36
72 Row space of a matrix 04:26
73 Null space and left null space of a matrix 14:40
74 Column/left-null and row/null spaces are orthogonal 10:48
75 Dimensions of column/row/null spaces 08:11
76 Example of the four subspaces 11:10
77 More on Ax=b and Ax=0 07:53
78 Systems of equations: algebra and geometry 19:40
79 Converting systems of equations to matrix equations 04:24
80 Gaussian elimination 14:43
81 Echelon form and pivots 07:22
82 Reduced row echelon form 18:30
83 Code challenge: RREF of matrices with different sizes and ranks 12:17
84 Matrix spaces after row reduction 09:24
85 Determinant: concept and applications 06:00
86 Determinant of a 2x2 matrix 07:04
87 Code challenge: determinant of small and large singular matrices 11:08
88 Determinant of a 3x3 matrix 13:14
89 Code challenge: large matrices with row exchanges 06:33
90 Find matrix values for a given determinant 04:52
91 Code challenge: determinant of shifted matrices 18:28
92 Code challenge: determinant of matrix product 10:38
93 Matrix inverse: Concept and applications 12:41
94 Computing the inverse in code 06:32
95 Inverse of a 2x2 matrix 07:56
96 The MCA algorithm to compute the inverse 13:59
97 Code challenge: Implement the MCA algorithm!! 18:40
98 Computing the inverse via row reduction 16:41
99 Code challenge: inverse of a diagonal matrix 10:51
100 Left inverse and right inverse 10:15
101 One-sided inverses in code 12:41
102 Proof: the inverse is unique 03:17
103 Pseudo-inverse, part 1 11:35
104 Code challenge: pseudoinverse of invertible matrices 06:03
105 Projections in R^2 10:00
106 Projections in R^N 15:25
107 Orthogonal and parallel vector components 12:39
108 Code challenge: decompose vector to orthogonal components 16:41
109 Orthogonal matrices 12:03
110 Gram-Schmidt procedure 12:44
111 QR decomposition 21:00
112 Code challenge: Gram-Schmidt algorithm 20:36
113 Matrix inverse via QR decomposition 01:46
114 Code challenge: Inverse via QR 14:20
115 Code challenge: Prove and demonstrate the Sherman-Morrison inverse 17:27
116 Code challenge: A^TA = R^TR 06:01
117 Introduction to least-squares 13:13
118 Least-squares via left inverse 10:08
119 Least-squares via orthogonal projection 09:19
120 Least-squares via row-reduction 18:21
121 Model-predicted values and residuals 07:00
122 Least-squares application 1 18:47
123 Least-squares application 2 29:41
124 Code challenge: Least-squares via QR decomposition 10:11
125 What are eigenvalues and eigenvectors? 12:53
126 Finding eigenvalues 20:44
127 Shortcut for eigenvalues of a 2x2 matrix 02:54
128 Code challenge: eigenvalues of diagonal and triangular matrices 14:25
129 Code challenge: eigenvalues of random matrices 11:05
130 Finding eigenvectors 15:57
131 Eigendecomposition by hand: two examples 09:28
132 Diagonalization 14:31
133 Matrix powers via diagonalization 20:37
134 Code challenge: eigendecomposition of matrix differences 18:15
135 Eigenvectors of distinct eigenvalues 08:15
136 Eigenvectors of repeated eigenvalues 12:16
137 Eigendecomposition of symmetric matrices 14:04
138 Eigenlayers of a matrix 07:20
139 Code challenge: reconstruct a matrix from eigenlayers 20:11
140 Eigendecomposition of singular matrices 05:00
141 Code challenge: trace and determinant, eigenvalues sum and product 10:57
142 Generalized eigendecomposition 12:31
143 Code challenge: GED in small and large matrices 21:10
144 Singular value decomposition (SVD) 18:41
145 Code challenge: SVD vs. eigendecomposition for square symmetric matrices 24:32
146 Relation between singular values and eigenvalues 13:04
147 Code challenge: U from eigendecomposition of A^TA 18:24
148 Code challenge: A^TA, Av, and singular vectors 14:34
149 SVD and the four subspaces 07:35
150 Spectral theory of matrices 21:57
151 SVD for low-rank approximations 16:43
152 Convert singular values to percent variance 15:26
153 Code challenge: When is UV^T valid, what is its norm, and is it orthogonal? 12:04
154 SVD, matrix inverse, and pseudoinverse 13:30
155 Condition number of a matrix 12:48
156 Code challenge: Create matrix with desired condition number 15:09
157 The quadratic form in algebra 15:28
158 The quadratic form in geometry 15:36
159 The normalized quadratic form 06:36
160 Code challenge: Visualize the normalized quadratic form 16:21
161 Eigenvectors and the quadratic form surface 06:18
162 Application of the normalized quadratic form: PCA 29:02
163 Quadratic form of generalized eigendecomposition 17:34
164 Matrix definiteness, geometry, and eigenvalues 12:55
165 Proof: A^TA is always positive (semi)definite 06:52
166 Proof: Eigenvalues and matrix definiteness 07:16

Similar courses to Complete linear algebra: theory and implementation

Complete Machine Learning and Data Science: Zero to Mastery
Python & LeetCode | The Ultimate Interview BootCamp
Django Masterclass : Build Web Apps With Python & Django
Python 3: Deep Dive (Part 4 - OOP)

Python 3: Deep Dive (Part 4 - OOP)udemy

Duration 35 hours 15 minutes 32 seconds
Build Fast Masterclass

Build Fast MasterclassBuildFast Academy

Duration 7 hours 22 minutes 11 seconds
Django 2.1 & Python | The Ultimate Web Development Bootcamp
100 Days of Code: The Complete Python Pro Bootcamp

100 Days of Code: The Complete Python Pro Bootcampudemy

Duration 54 hours 16 minutes 30 seconds
Compilers, Interpreters and Formal Languages

Compilers, Interpreters and Formal LanguagesGustavo Pezzi

Duration 12 hours 3 minutes 54 seconds
Build a Python REST API with the Django Rest Framework
The Ultimate Django Series: Part 2