Complete linear algebra: theory and implementation in code
- Descrição
- Currículo
- FAQ
- Revisões
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.
-
Linear algebra is a prerequisite of machine learning and artificial intelligence (A.I.).
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.
So what are you waiting for??
Watch the course introductory video and free sample videos to learn more about the contents of this course and about my teaching style. If you are unsure if this course is right for you and want to learn more, feel free to contact with me questions before you sign up.
I hope to see you soon in the course!
Mike
-
7
Algebraic and geometric interpretations of vectorsLearn two ways of interpreting vectors (this is the "algebraic-geometric dualist perspective" in linear algebra).
-
8
Vector addition and subtraction
How to do basic arithmetic with vectors.
-
9
Vector-scalar multiplication
Multiply a vector by a number, and learn why "scalars" are called scalars.
-
10
Vector-vector multiplication: the dot product
Arguably the most important and fundamental computation in all of linear algebra!
-
11
Dot product properties: associative, distributive, commutative
Learn several important properties of the vector dot product.
-
12
Code challenge: dot products with matrix columns
Use a for-loop to compute dot products between corresponding columns.
-
13
Code challenge: is the dot product commutative?
Is the dot product commutative? Use a computer to find out!
-
14
Vector length
Learn to compute the length of a vector.
-
15
Vector length in MATLABFind the coding bug!
-
16
Vector length in Python
Which of the following lines of Python code contains an error in computing the length of vector v?
-
17
Dot product geometry: sign and orthogonality
How to interpret the sign of a dot product from a geometric perspective.
-
18
Vector orthogonality
Determine whether two vectors in R3 are orthogonal.
-
19
Code challenge: Cauchy-Schwarz inequality
-
20
Relative vector angles
Use the dot product sign to infer geometric relationships.
-
21
Code challenge: dot product sign and scalar multiplication
Implement what you learned in code!
-
22
Vector Hadamard multiplication
Learn the "sensible way" to multiply two vectors.
-
23
Outer product
Create a matrix from two vectors using the outer product.
-
24
Vector cross product
The special multiplication for 3-D vectors.
-
25
Vectors with complex numbers
Learn the basics of complex numbers and complex vectors.
-
26
Hermitian transpose (a.k.a. conjugate transpose)
If you ever work with complex numbers in linear algebra, you need to know about the Hermitian!
-
27
Interpreting and creating unit vectors
"Normalize" a vector by giving it length=1.
-
28
Code challenge: dot products with unit vectors
-
29
Dimensions and fields in linear algebra
Important linear algebra terminology.
-
30
Subspaces
A subspace is an important concept in linear algebra that is fundamental for many other topics.
-
31
Subspaces vs. subsets
Two very different but easily confused topics.
-
32
Span
Learn the algebraic and geometric interpretations of a span of a set of vectors.
-
33
In the span?
Determine whether a vector is in the span of a set of vectors.
-
34
Linear independence
The linear algebra declaration of linear independence!
-
35
Basis
Combine independence and basis into one concept.
-
36
Matrix terminology and dimensionality
Learn the basic terminology of matrices.
-
37
Matrix sizes and dimensionality
-
38
A zoo of matrices
Many matrices are given special names, here are some of them.
-
39
Can the matrices be concatenated?
-
40
Matrix addition and subtraction
Basic arithmetic with matrices.
-
41
Matrix-scalar multiplication
Multiply a matrix by a number.
-
42
Code challenge: is matrix-scalar multiplication a linear operation?
Use computers to test whether u(A+M) = uA+uM
-
43
Transpose
Flipping off a matrix or vector is actually a good thing in linear algebra.
-
44
Complex matrices
What you learned with complex vectors applies to matrices.
-
45
Addition, equality, and transpose
True or false
-
46
Diagonal and trace
How to work with the diagonal elements of a matrix.
-
47
Code challenge: linearity of trace
Apply your knowledge to learn a new concept in linear algebra.
-
48
Broadcasting matrix arithmetic
-
49
Introduction to standard matrix multiplication
Matrix multiplication gets its own introduction.
-
50
Four ways to think about matrix multiplication
Strange but true: There are four different ways to think about matrix multiplication.
-
51
Code challenge: matrix multiplication by layering
Implement matrix multiplication in code.
-
52
Matrix multiplication with a diagonal matrix
Diagonal matrices are convenient for many reasons, including simplicity of multiplication.
-
53
Order-of-operations on matrices
Learn the "LIVE EVIL" rule!
-
54
Matrix-vector multiplication
Key properties of matrix-vector multiplication.
-
55
Find the missing value!
Find the value for * that makes the equation valid.
-
56
2D transformation matrices
A geometric interpretation of matrix-vector multiplication.
-
57
Code challenge: Pure and impure rotation matrices
-
58
Code challenge: Geometric transformations via matrix multiplications
Also, gain new insight into the meaning of singular values!
-
59
Additive and multiplicative matrix identities
Two key matrix identities lead to the zero matrix and the identity matrix.
-
60
Additive and multiplicative symmetric matrices
Learn how to create symmetric matrices.
-
61
Hadamard (element-wise) multiplication
Yet another way to multiply matrices.
-
62
Matrix operation equality
Determine whether two operations on matrices give identical results.
-
63
Code challenge: symmetry of combined symmetric matrices
Use code to learn an important concept in linear algebra.
-
64
Multiplication of two symmetric matrices
Is the product of two symmetric matrices symmetric? Find out!
-
65
Code challenge: standard and Hadamard multiplication for diagonal matrices
Use code to learn a special property of multiplication with diagonal matrices.
-
66
Code challenge: Fourier transform via matrix multiplication!
Create a Fourier matrix and implement the Fourier transform.
-
67
Frobenius dot product
The Frobenius dot product is used often in statistics and machine learning.
-
68
Matrix norms
Learn several commonly used matrix norms.
-
69
Code challenge: conditions for self-adjoint
-
70
Code challenge: The matrix asymmetry index
-
71
What about matrix division?
Conceptual and implementational aspects of matrix division.
-
72
Rank: concepts, terms, and applications
Learn the key properies and uses of matrix rank.
-
73
Maximum possible rank.
-
74
Computing rank: theory and practice
Learn the distinction between rank in a first-year course vs. rank in real-world applications.
-
75
Rank of added and multiplied matrices
Upper bounds of the ranks of added and multiplied matrices.
-
76
What's the maximum possible rank?
Test your knowledge of the rank of summed and multiplied matrices.
-
77
Code challenge: reduced-rank matrix via multiplication
Create an MxN matrix with rank r.
-
78
Code challenge: scalar multiplication and rank
Does scalar multiplication change the rank of a matrix? Test your hypothesis in code!
-
79
Rank of A^TA and AA^T
Rank of our favorite type of matrix: A^TA.
-
80
Code challenge: rank of multiplied and summed matrices
Use code to confirm your theoretical knowledge.
-
81
Making a matrix full-rank by "shifting"
Transform a rank-deficient to a full-rank matrix using this one simple trick!
-
82
Code challenge: is this vector in the span of this set?
Use code to test span.
-
83
Course tangent: self-accountability in online learning
-
84
Column space of a matrix
Apply a concept you've already learned (span) to a new domain.
-
85
Column space, visualized in code
The column space of a matrix, visualized in MATLAB.
-
86
Row space of a matrix
Really, it's the same as the column space, but transposed.
-
87
Null space and left null space of a matrix
Learn how to interpret and find the "null space" of a matrix.
-
88
Column/left-null and row/null spaces are orthogonal
Learn some interesting features of matrix spaces.
-
89
Dimensions of column/row/null spaces
See how the puzzle pieces of matrix spaces all fit together.
-
90
Example of the four subspaces
See an example of extracting bases for the four subspaces of a matrix.
-
91
More on Ax=b and Ax=0
These equations look simple, but they are really important for applied linear algebra.
