Linear Algebra With Applications

Comprehensive linear algebra course that covers all the important topics from theory to applications

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 forumUdemy 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

Overview

Section 1: Introduction

Lecture 1 Course introduction

Lecture 2 Efficient way to study this course

Section 2: Systems of Linear Equations and Matrices

Lecture 3 Introduction to linear equations

Lecture 4 General form of systems of linear equations

Lecture 5 Solutions to linear systems with two unknowns

Lecture 6 Solutions to linear systems with three unknowns

Lecture 7 Worked examples on solving linear systems

Lecture 8 Augmented matrices

Lecture 9 Row operations using augmented matrices

Lecture 10 Row echelon forms

Lecture 11 Worked examples on row echelon forms

Lecture 12 Gauss-Jordan vs Gaussian elimination

Lecture 13 Homogeneous linear systems

Lecture 14 Gaussian elimination with back substitution

Lecture 15 Matrix and vector notation, size of a matrix

Lecture 16 Matrix operations (addition, subtraction, equality, scalar product, trace)

Lecture 17 Matrix multiplication

Lecture 18 Partitioned matrices

Lecture 19 Matrix products as linear combinations

Lecture 20 Matrix transpose

Lecture 21 Computing the inverse of a matrix

Lecture 22 Inverse of a product of matrices

Lecture 23 Powers of matrices

Lecture 24 Inverse of a 3x3 matrix by Gauss-Jordan elimination

Lecture 25 Solving linear systems by matrix inversion

Lecture 26 Diagonal matrices, inverse and powers

Lecture 27 Triangular matrices, inverse and transpose

Lecture 28 Symmetric matrices, inverse and transpose

Section 3: Matrix Determinants and Inverse

Lecture 29 Determinant of a matrix

Lecture 30 Computing the determinant by Gaussian elimination

Lecture 31 Computing the inverse of a matrix using the adjoint matrix

Lecture 32 Cramer's rule

Section 4: Extending Vectors from 3-Space to n-Space

Lecture 33 Vectors in 2D and 3D space

Lecture 34 Vectors in n-space

Lecture 35 Worked example on vectors in n-space

Lecture 36 Norm of a vector in n-space

Lecture 37 Standard unit vectors in n-space

Lecture 38 Dot product

Lecture 39 Dot product in component form

Lecture 40 Dot product in n-space

Lecture 41 Properties of the dot product in n-space

Lecture 42 Cauchy-Schwarz inequality, dot product angle in n-space

Lecture 43 Triangle inequality

Lecture 44 Dot product by matrix multiplication

Lecture 45 Orthogonal vectors and the dot product

Lecture 46 Applying the dot product to the geometry of a line and plane

Lecture 47 Vector projection using the dot product

Lecture 48 Cross product

Lecture 49 Geometric interpretation of the cross product

Lecture 50 Properties of the cross product

Lecture 51 Scalar triple product

Lecture 52 Volume of a parallelepiped and the scalar triple product

Section 5: Real Vector Spaces

Lecture 53 Real vector spaces

Lecture 54 Worked example on real vector spaces (1 of 3)

Lecture 55 Worked example on real vector spaces (2 of 3)

Lecture 56 Worked example on real vector spaces (3 of 3)

Lecture 57 Vector subspaces

Lecture 58 Subspaces of common vector spaces

Lecture 59 Subsets that are not vector subspaces

Lecture 60 Linear combinations and span of a vector subspace

Lecture 61 Linearly independent sets

Lecture 62 Linearly independent vectors

Lecture 63 Worked example on linearly independent vectors

Lecture 64 Basis for a vector space

Lecture 65 Vector spaces with no basis

Lecture 66 Basis representation of vectors and uniqueness

Lecture 67 Worked example on basis for a vector space (1 of 2)

Lecture 68 Worked example on basis for a vector space and coordinate vectors (2 of 2)

Lecture 69 Dimension of a vector space

Lecture 70 Verifying a basis for a vector space using the dimension

Lecture 71 Dimension of a vector subspace

Lecture 72 Worked example on the dimension of a vector space

Lecture 73 Coordinate vectors and mapping

Lecture 74 Change of basis

Lecture 75 Transition matrix

Lecture 76 Determining the transition matrix for R^n space

Section 6: Fundamental Matrix Spaces

Lecture 77 Row, column and null space

Lecture 78 Consistency of Ax=b

Lecture 79 Superposition of solutions for Ax=b and Ax=0

Lecture 80 Effect of row operations on row, column and null space

Lecture 81 Basis for row and column space

Lecture 82 More on the basis for a column space

Lecture 83 Finding the basis from a set of vectors

Lecture 84 Rank and nullity of a matrix

Lecture 85 Number of parameters in the general solution for Ax=b

Lecture 86 Overdetermined systems

Lecture 87 Underdetermined systems

Lecture 88 Fundamental matrix spaces

Lecture 89 Orthogonal complements

Section 7: Matrix Transformations, Operators

Lecture 90 Quick revision on functions

Lecture 91 Transformations and vector spaces

Lecture 92 Matrix transformations

Lecture 93 Properties of matrix transformations

Lecture 94 Finding standard matrices

Lecture 95 Worked example on matrix transformation operators (1 of 4)

Lecture 96 Worked example on matrix transformation operators (2 of 4)

Lecture 97 Worked example on matrix transformation operators (3 of 4)

Lecture 98 Worked example on matrix transformation operators (4 of 4)

Lecture 99 Compositions of matrix transformations

Lecture 100 One-to-one transformations

Lecture 101 Inverse of a matrix operator

Lecture 102 Worked example on matrix transformations

Section 8: Eigenvalues and Eigenvectors, Complex Vector Spaces

Lecture 103 Eigenvalues and eigenvectors, definition

Lecture 104 Calculating eigenvalues

Lecture 105 Eigenvectors and eigenspaces

Lecture 106 Geometric interpretation of eigenvalues, eigenvectors and eigenspaces

Lecture 107 Eigenvalues and eigenvectors of matrix powers

Lecture 108 Similar matrices

Lecture 109 Diaogonalisability of matrices

Lecture 110 Matrix powers

Lecture 111 Geometric and algebraic multiplicity

Lecture 112 Revision on complex numbers

Lecture 113 Complex eigenvalues

Lecture 114 Complex vector spaces

Lecture 115 Complex matrices

Lecture 116 Complex Euclidean inner product

Lecture 117 Geometric interpretation of complex matrices

Section 9: Inner Product Spaces

Lecture 118 Inner product space

Lecture 119 Norm and distance

Lecture 120 Matrix inner products

Lecture 121 Algebraic properties of inner products

Lecture 122 Worked example on inner product spaces (1 of 2)

Lecture 123 Worked example on inner product spaces (2 of 2)

Lecture 124 Angle between vectors in inner product spaces

Lecture 125 Length and distance

Lecture 126 Orthogonality

Lecture 127 Orthogonal complements

Lecture 128 Orthogonal sets and linear independence

Lecture 129 Orthonormal basis and coordinate vectors

Lecture 130 Orthogonal projection

Lecture 131 Gram-Schmidt process for obtaining an orthonormal basis

Section 10: Orthogonal Diagonalisation, Symmetric Matrices and Quadratic Forms

Lecture 132 Orthogonal matrices

Lecture 133 Properties of the inverse of an orthogonal matrix

Lecture 134 Orthogonal linear operators

Lecture 135 Orthonormal basis, properties

Lecture 136 Change of orthonormal basis

Lecture 137 Orthogonal diagonalisability

Lecture 138 Symmetric matrices and their properties

Lecture 139 Orthogonal diagonalisation process

Lecture 140 Spectral decomposition

Lecture 141 Quadratic forms, definition

Lecture 142 Principle axes theorem

Lecture 143 Conics and quadratic forms

Lecture 144 Identifying rotated conics

Lecture 145 Positive definite quadratic forms

Lecture 146 Conjugate transpose of complex matrices

Lecture 147 Hermitian matrices

Lecture 148 Unitary matrices

Lecture 149 Unitary diagonalisability

Lecture 150 Unitary diagonalisation process

Lecture 151 Skew symmetric and skew Hermitian matrices

Section 11: Numerical Methods

Lecture 152 LU solution method

Lecture 153 LU decomposition

Lecture 154 Simplified LU decomposition

Lecture 155 LDU factorisation

Lecture 156 Power method, estimating eigenvalues and eigenvectors by iteration

Lecture 157 Geometric interpretation of the power method

Lecture 158 Power method with Euclidean scaling

Lecture 159 Power method with maximum entry scaling

Lecture 160 Rate of convergence of the power method

Lecture 161 Stopping condition for the power method

Lecture 162 Closest approximation to a vector in a subspace

Lecture 163 Least squares approximation

Lecture 164 Least squares solutions (LSS)

Lecture 165 Uniqueness of LSS and orthogonal projection onto the column space

Lecture 166 Projection transformation

Lecture 167 Matrix decompositions

Lecture 168 Properties of the A^T.A matrix

Lecture 169 Singular values

Lecture 170 Singular value decomposition

Lecture 171 QR decomposition

Lecture 172 Worked example on QR decomposition

Lecture 173 Gauss-Seidel iteration

Lecture 174 General form the Gauss-Seidel method

Lecture 175 Jacobi iteration

Section 12: Applications of Linear Algebra

Lecture 176 Balancing chemical equations

Lecture 177 Polynomial interpolation for approximating integrals

Lecture 178 Worked example on polynomial interpolation

Lecture 179 Linear systems of ordinary differential equations (ODE)

Lecture 180 Solution to linear systems of ODEs by diagonalisation

Lecture 181 Worked example on solving linear systems of ODEs

Lecture 182 Linear regression using the least squares method

Lecture 183 Worked example on linear regression

Lecture 184 Approximating functions and the mean squared error

Lecture 185 Least squares approximation

Lecture 186 Fourier series

Lecture 187 Function approximation problem

Lecture 188 Worked example on 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