Master Linear Algebra (Matrices, Vector Spaces, Numerical)

Learn and understand all key topics from Linear Algebra with intuitive geometric interpretations and practice examples

What you'll learn
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

Requirements
Basic algebra
Minimal Calculus 2 (integration and ordinary differential equations) knowledge for some of the applications (last section)

Description
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 classAny person who is interested in mathematics and/or needs a refresher courseAny 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 onAt 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 toFriendly Q&A forum50 targeted practice problems (more will be added continuously)Udemy Certificate of Completion30-day money back guaranteeThe course covers the following core units and topics of Linear Algebra:1) Systems of Linear Equations and Matricesa) Introduction to linear equations and general form of linear systemsb) Solutions to linear systems with two or three unknownsc) Augmented matrices and row operationsd) Row echelon formse) Gauss-Jordan vs Gaussian elimination (with back substitution)f) Homogeneous linear systemsg) Matrix and vector notation, size, and matrix operationsh) Partitioned matricesi) Inverse of a matrix or product of matrices and solving linear systems by matrix inversionj) Diagonal, triangular and symmetric matrices, and their inverse, transpose and powers2) Matrix Determinants and Inversea) Determinant of a matrix using minor matrices and Gaussian eliminationb) Computing the inverse of a matrix using the adjoint matrixc) Cramer's rule3) Extending Vectors from 3-Space to n-Spacea) Vectors in 2D and 3D spaceb) Vectors in n-spacec) Norm of a vector in n-space and the standard unit vectorsd) Dot product in n-spacee) Orthogonality and projection using the dot productf) Cross product, scalar triple product, area and volume4) Real Vector Spacesa) Real vector spacesb) Vector subspaces, span, linear combinationsc) Linearly independent vectors and linear independenced) Basis for a vector space and coordinate vectorse) Dimension of a vector spacef) Change of basis, coordinate vectors, mapping and transition matrix5) Fundamental Matrix Spacesa) Row, column and null spaceb) Consistency of linear systems and superposition of solutionsc) Effect of row operations on row, column and null spaced) Basis for row and column spacee) Rank and nullity of a matrixf) Overdetermined and underdetermined systemsg) Fundamental matrix spacesh) Orthogonal complements6) Matrix Transformations, Operatorsa) Matrix transformations and their propertiesb) Finding standard matricesc) Operators, including projection, reflection, rotation and sheard) Compositions of matrix transformationse) One-to-one transformations and the inverse of a matrix operator7) Eigenvalues and Eigenvectors, Complex Vector Spacesa) Eigenvalues, eigenvectors and eigenspacesb) Similar matrices and diagonalisationc) Complex vector spaces, eigenvalues, eigenvectors and Euclidean inner product8) Inner Product Spacesa) Inner product spaces, norm and distance, matrix inner productsb) Orthogonalityc) Gram-Schmidt process for finding an orthonormal basis from an orthogonal set9) Orthogonal Diagonalisation, Symmetric Matrices and Quadratic Formsa) Orthogonal matricesb) Orthogonal diagonalisation and spectral decompositionc) Quadratic forms, conic sections, positive definitenessd) Conjugate transpose, diagonalisation of Hermitian and unitary matrices10) Numerical Methodsa) LU and LDU decomposition or factorisationb) Power method for estimating eigenvalues and eigenvectors using iterationc) Least squares approximationd) Singular value decompositione) QR decompositionf) Gauss-Seidel and Jacobi iteration11) Applicationsa) Balancing chemical equationsb) Approximating integrals by polynomial interpolationc) Solving linear systems of ODEs by diagonalisationd) Linear regression using the least squares methode) Approximating functions using the least squares method and Fourier series

Who this course is 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