Overview
Solve systems of linear equations using matrices and various methods like Gaussian vs Gauss-Jordan Elimination, row echelon forms, row operations, Find the deteminant and inverse of a matrix, and apply Cramer's rule, Vectors and their operations in 2D and 3D space, including addition, scalar multiplication, subtraction, representation in coordinate systems, position vectors, Extend vectors to n-space, including norm, standard unit vectors, dot product, angle using the Cauchy-Schwarz inequality, Orthogonality and projection using the dot product, geometric interpretation of the cross product and triple scalar product, Real vector spaces, subspaces, linear combinations and span, linear independence, basis, dimension, change of basis, computing the transition matrix, Row space column space and null space, basis and effect of row operations on these spaces, Rank, nullity, fundamental matrix spaces, overdetermined and underdetermined systems, orthogonal complements, Matrix transformations and their properties, finding standard matrices, compositions, one-to-one, Eigenvalues, eigenvectors, eigenspaces, geometric interpretation, matrix powers, diagonalising similar matrices, geometric and algebraic multiplicity, Complex vector spaces, eigenvalues, eigenvectors, matrices and inner product, geometric interpretation, Inner product spaces, orthogonality, Gram-Schmidt process and orthonormal basis, orthogonal projection, Orthogonal diagonalisation, symmetric matrices and spectral decomposition, Quadratic forms, principal axes theorem, conics, positive definiteness, Diagonalisation of complex matrices, Hermitian and unitary matrices, skew symmetric and sew Hermitian matrices, Direct/iterative numerical methods, including LU and LDU factorisation, power method, least squares, singular value and QR decomposition, Gauss-Seidel iteration, Applications, including balancing chemical equations, polynomial interpolation, solving systems of ODEs, linear regression, and approximating functions
Linear algebra students who want to be at the top of their class, Any person who is interested in mathematics and/or needs a refresher course, Any person who is undertaking a discipline that requires linear algebra, including science, physics and engineering, graphics and games programming, finite element analysis, machine learning, big data analysis, economics, finance and so on
Basic algebra, Minimal Calculus 2 (integration and ordinary differential equations) knowledge for some of the applications (last section)
This course is designed to make learning Linear Algebra easy. It is well-arranged into targeted sections of focused lectures and extensive worked examples to give you a solid foundation in the key topics from theory to applications.
The course is ideal for:
Linear algebra students who want to be at the top of their class
Any person who is interested in mathematics and/or needs a refresher course
Any person who is undertaking a discipline that requires linear algebra, including science, physics and engineering, graphics and games programming, finite element analysis, machine learning, big data analysis, economics, finance and so on
At the end of this course, you will have a strong foundation in one of the most disciplines in Applied Mathematics, which you will definitely come across if you are from a science, computer science, engineering, economics or finance background.
I welcome any questions and provide a friendly Q&A forum where I aim to respond to you in a timely manner.
Enrol today and you will get:
Lifetime access to refer back to the course whenever you need to
Friendly Q&A forum
Udemy Certificate of Completion
30-day money back guarantee
The course covers the following core units and topics of Linear Algebra:
1) Systems of Linear Equations and Matrices
a) Introduction to linear equations and general form of linear systems
b) Solutions to linear systems with two or three unknowns
c) Augmented matrices and row operations
d) Row echelon forms
e) Gauss-Jordan vs Gaussian elimination (with back substitution)
f) Homogeneous linear systems
g) Matrix and vector notation, size, and matrix operations
h) Partitioned matrices
i) Inverse of a matrix or product of matrices and solving linear systems by matrix inversion
j) Diagonal, triangular and symmetric matrices, and their inverse, transpose and powers
2) Matrix Determinants and Inverse
a) Determinant of a matrix using minor matrices and Gaussian elimination
b) Computing the inverse of a matrix using the adjoint matrix
c) Cramer's rule
3) Extending Vectors from 3-Space to n-Space
a) Vectors in 2D and 3D space
b) Vectors in n-space
c) Norm of a vector in n-space and the standard unit vectors
d) Dot product in n-space
e) Orthogonality and projection using the dot product
f) Cross product, scalar triple product, area and volume
4) Real Vector Spaces
a) Real vector spaces
b) Vector subspaces, span, linear combinations
c) Linearly independent vectors and linear independence
d) Basis for a vector space and coordinate vectors
e) Dimension of a vector space
f) Change of basis, coordinate vectors, mapping and transition matrix
5) Fundamental Matrix Spaces
a) Row, column and null space
b) Consistency of linear systems and superposition of solutions
c) Effect of row operations on row, column and null space
d) Basis for row and column space
e) Rank and nullity of a matrix
f) Overdetermined and underdetermined systems
g) Fundamental matrix spaces
h) Orthogonal complements
6) Matrix Transformations, Operators
a) Matrix transformations and their properties
b) Finding standard matrices
c) Operators, including projection, reflection, rotation and shear
d) Compositions of matrix transformations
e) One-to-one transformations and the inverse of a matrix operator
7) Eigenvalues and Eigenvectors, Complex Vector Spaces
a) Eigenvalues, eigenvectors and eigenspaces
b) Similar matrices and diagonalisation
c) Complex vector spaces, eigenvalues, eigenvectors and Euclidean inner product
8) Inner Product Spaces
a) Inner product spaces, norm and distance, matrix inner products
b) Orthogonality
c) Gram-Schmidt process for finding an orthonormal basis from an orthogonal set
9) Orthogonal Diagonalisation, Symmetric Matrices and Quadratic Forms
a) Orthogonal matrices
b) Orthogonal diagonalisation and spectral decomposition
c) Quadratic forms, conic sections, positive definiteness
d) Conjugate transpose, diagonalisation of Hermitian and unitary matrices
10) Numerical Methods
a) LU and LDU decomposition or factorisation
b) Power method for estimating eigenvalues and eigenvectors using iteration
c) Least squares approximation
d) Singular value decomposition
e) QR decomposition
f) Gauss-Seidel and Jacobi iteration
11) Applications
a) Balancing chemical equations
b) Approximating integrals by polynomial interpolation
c) Solving linear systems of ODEs by diagonalisation
d) Linear regression using the least squares method
e) Approximating functions using the least squares method and Fourier series
DrOf Eng
I am teaching Math, Science and Engineering subjects. Throughout my last 10 years of tutoring students, I have tested effective teaching methods that have not only helped students to receive excellent grades but understand concepts through problem-based learning.
I have a Bachelor of Information Systems and Masters/PhD in Structural Engineering.
I currently create Math, Science and Engineering videos to help students access effective and affordable education.