IIRF Online > Teaching & Academics > Math > Linear Algebra > Complete linear algebra: theory and implementation in code
Complete linear algebra: theory and implementation in code by Udemy
Learn concepts in linear algebra and matrix analysis, and implement them in MATLAB and Python.
Course Highlights
- 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
- The math underlying most of AI (artificial intelligence)
Skills you will learn!
Curriculum
5 Topics
What is linear algebra?
Linear algebra applications
An enticing start to a linear algebra course!
How best to learn from this course
Maximizing your Udemy experience
1 Topic
How to download and use course materials
29 Topics
Algebraic and geometric interpretations of vectors
Vector addition and subtraction
Vector-scalar multiplication
Vector-vector multiplication: the dot product
Dot product properties: associative distributive commutative
Code challenge: dot products with matrix columns
Code challenge: is the dot product commutative?
Vector length
Vector length in MATLAB
Vector length in Python
Dot product geometry: sign and orthogonality
Vector orthogonality
Code challenge: Cauchy-Schwarz inequality
Relative vector angles
Code challenge: dot product sign and scalar multiplication
Vector Hadamard multiplication
Outer product
Vector cross product
Vectors with complex numbers
Hermitian transpose (a.k.a. conjugate transpose)
Interpreting and creating unit vectors
Code challenge: dot products with unit vectors
Dimensions and fields in linear algebra
Subspaces
Subspaces vs. subsets
Span
In the span?
Linear independence
Basis
13 Topics
Matrix terminology and dimensionality
Matrix sizes and dimensionality
A zoo of matrices
Can the matrices be concatenated?
Matrix addition and subtraction
Matrix-scalar multiplication
Code challenge: is matrix-scalar multiplication a linear operation?
Transpose
Complex matrices
Addition equality and transpose
Diagonal and trace
Code challenge: linearity of trace
Broadcasting matrix arithmetic
23 Topics
Introduction to standard matrix multiplication
Four ways to think about matrix multiplication
Code challenge: matrix multiplication by layering
Matrix multiplication with a diagonal matrix
Order-of-operations on matrices
Matrix-vector multiplication
Find the missing value!
2D transformation matrices
Code challenge: Pure and impure rotation matrices
Code challenge: Geometric transformations via matrix multiplications
Additive and multiplicative matrix identities
Additive and multiplicative symmetric matrices
Hadamard (element-wise) multiplication
Matrix operation equality
Code challenge: symmetry of combined symmetric matrices
Multiplication of two symmetric matrices
Code challenge: standard and Hadamard multiplication for diagonal matrices
Code challenge: Fourier transform via matrix multiplication!
Frobenius dot product
Matrix norms
Code challenge: conditions for self-adjoint
Code challenge: The matrix asymmetry index
What about matrix division?
12 Topics
Rank: concepts terms and applications
Maximum possible rank.
Computing rank: theory and practice
Rank of added and multiplied matrices
What's the maximum possible rank?
Code challenge: reduced-rank matrix via multiplication
Code challenge: scalar multiplication and rank
Rank of A^TA and AA^T
Code challenge: rank of multiplied and summed matrices
Making a matrix full-rank by "shifting"
Code challenge: is this vector in the span of this set?
Course tangent: self-accountability in online learning
8 Topics
Column space of a matrix
Column space visualized in code
Row space of a matrix
Null space and left null space of a matrix
Column/left-null and row/null spaces are orthogonal
Dimensions of column/row/null spaces
Example of the four subspaces
More on Ax=b and Ax=0
7 Topics
Systems of equations: algebra and geometry
Converting systems of equations to matrix equations
Gaussian elimination
Echelon form and pivots
Reduced row echelon form
Code challenge: RREF of matrices with different sizes and ranks
Matrix spaces after row reduction
8 Topics
Determinant: concept and applications
Determinant of a 2x2 matrix
Code challenge: determinant of small and large singular matrices
Determinant of a 3x3 matrix
Code challenge: large matrices with row exchanges
Find matrix values for a given determinant
Code challenge: determinant of shifted matrices
Code challenge: determinant of matrix product
13 Topics
Matrix inverse: Concept and applications
Computing the inverse in code
Inverse of a 2x2 matrix
The MCA algorithm to compute the inverse
Code challenge: Implement the MCA algorithm!!
Computing the inverse via row reduction
Code challenge: inverse of a diagonal matrix
Left inverse and right inverse
One-sided inverses in code
Proof: the inverse is unique
Pseudo-inverse part 1
Code challenge: pseudoinverse of invertible matrices
Why should you avoid the inverse?
12 Topics
Projections in R^2
Projections in R^N
Orthogonal and parallel vector components
Code challenge: decompose vector to orthogonal components
Orthogonal matrices
Gram-Schmidt procedure
QR decomposition
Code challenge: Gram-Schmidt algorithm
Matrix inverse via QR decomposition
Code challenge: Inverse via QR
Code challenge: Prove and demonstrate the Sherman-Morrison inverse
Code challenge: A^TA = R^TR
8 Topics
Introduction to least-squares
Least-squares via left inverse
Least-squares via orthogonal projection
Least-squares via row-reduction
Model-predicted values and residuals
Least-squares application 1
Least-squares application 2
Code challenge: Least-squares via QR decomposition
19 Topics
What are eigenvalues and eigenvectors?
Finding eigenvalues
Shortcut for eigenvalues of a 2x2 matrix
Code challenge: eigenvalues of diagonal and triangular matrices
Code challenge: eigenvalues of random matrices
Finding eigenvectors
Eigendecomposition by hand: two examples
Diagonalization
Matrix powers via diagonalization
Code challenge: eigendecomposition of matrix differences
Eigenvectors of distinct eigenvalues
Eigenvectors of repeated eigenvalues
Eigendecomposition of symmetric matrices
Eigenlayers of a matrix
Code challenge: reconstruct a matrix from eigenlayers
Eigendecomposition of singular matrices
Code challenge: trace and determinant eigenvalues sum and product
Generalized eigendecomposition
Code challenge: GED in small and large matrices
16 Topics
Singular value decomposition (SVD)
Are these two expressions equal?
Code challenge: SVD vs. eigendecomposition for square symmetric matrices
Relation between singular values and eigenvalues
Code challenge: U from eigendecomposition of A^TA
SVD and the four subspaces
Spectral theory of matrices
SVD for low-rank approximations
Convert singular values to percent variance
Code challenge: When is UV^T valid what is its norm and is it orthogonal?
Singular values of an orthogonal matrix
SVD matrix inverse and pseudoinverse
SVD (pseudo)inverse and left-inverse
Condition number of a matrix
Code challenge: Create matrix with desired condition number
Code challenge: Why you avoid the inverse
10 Topics
The quadratic form in algebra
The quadratic form in geometry
The normalized quadratic form
Code challenge: Visualize the normalized quadratic form
Eigenvectors and the quadratic form surface
Application of the normalized quadratic form: PCA
Quadratic form of generalized eigendecomposition
Matrix definiteness geometry and eigenvalues
Proof: A^TA is always positive (semi)definite
Proof: Eigenvalues and matrix definiteness
1 Topic
Bonus lecture
Complete linear algebra: theory and implementation in code
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