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    Linear Algebra With Applications

    Posted By: ELK1nG
    Linear Algebra With Applications

    Linear Algebra With Applications
    Published 6/2025
    MP4 | Video: h264, 1280x720 | Audio: AAC, 44.1 KHz
    Language: English | Size: 4.56 GB | Duration: 14h 1m

    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