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    Linear Algebra and Geometry 3

    BY
    Udemy

    Study multiple product spaces, quadratics, and the essential principle and concepts associated with linear algebra and geometry 3 to solve complex equations.

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    Mode

    Online

    Fees

    ₹ 2699

    Quick Facts

    particular details
    Medium of instructions English
    Mode of learning Self study
    Mode of Delivery Video and Text Based

    Course overview

    The Linear Algebra and Geometry 3 certification course was developed by Hania Uscka-Wehlou, a Ph.D. and a university mathematics teacher, and is available on Udemy for those interested in learning the features of both geometry 3 and linear algebra. The Linear Algebra and Geometry 3 online course by Udemy focuses on teaching participants how to solve a mathematical issue using the principles of geometry 3 and linear algebra.

    Linear Algebra and Geometry 3 online classes feature 51 hours of prerecorded lectures as well as 316 downloadable resources on topics such as orthogonality, diagonalization, singular value decomposition, eigendecomposition, triangle inequality, matrices, vectors, quadratic forms, pseudoinverses, conic sections, and more. Participants will also receive a certificate recognizing their proficiency in the concepts of geometry 3 and linear algebra upon successful completion of the program.

    The highlights

    • Certificate of completion
    • Self-paced course
    • 51 hours of pre-recorded video content
    • 316 downloadable resources

    Program offerings

    • Online course
    • Learning resources. 30-day money-back guarantee
    • Unlimited access
    • Accessible on mobile devices and tv

    Course and certificate fees

    Fees information
    ₹ 2,699
    certificate availability

    Yes

    certificate providing authority

    Udemy

    Who it is for

    Teacher

    What you will learn

    Mathematical skill

    After completing the Linear Algebra and Geometry 3 online certification, participants will gain a detailed understanding of the mathematical concepts including geometry 3 and linear algebra. Participants will explore the theory associated with geometry and linear algebra including Pythagoras theorem, triangle inequality, Gram-Schmidt process, Cauchy-Schwarz inequality, and more. Participants will learn about principles associated with eigendecomposition, singular value decomposition, pseudoinverses, diagonalization, and orthogonality. Participants will also learn about matrices, quadratic forms, vectors, vector spaces, quadratic surfaces, conic sections, orthogonal bases, and orthonormal bases.

    The syllabus

    Introduction

    • Introduction

    Geometrical operators in the plane and in the 3-space

    • Eigendecomposition, recap
    • Eigendecomposition and operators
    • Problem 1: Line symmetry in the plane
    • Problem 2: Projection in the plane
    • Problem 3: Symmetry in the 3-space
    • Problem 4: Projection in the 3-space
    • Problem 5: Projection in the 3-space
    • Another formulation of eigendecomposition: Spectral decomposition
    • Powers of matrices: Two methods
    • Spectral decomposition, Problem 6
    • Spectral decomposition, Problem 7
    • Spectral decomposition, Geometrical illustration, Problem 8

    More problem solving; spaces different from R^n

    • Eigendecomposition, Problem 1
    • Eigendecomposition, Problem 2
    • Powers and roots, Problem 3
    • Powers and roots, Problem 4
    • In the space of polynomials, Problem 5
    • In the space of polynomials, Problem 6
    • In the space of matrices, Problem 7

    Intermezzo: isomorphic vector spaces

    • You wouldn’t see the difference
    • Different spaces with the same structure
    • More examples of isomorphic vector spaces
    • A necessary condition for isomorphic vector spaces
    • A necessary and sufficient condition for isomorphic vector spaces
    • Why you don’t see the difference
    • Isomorphic spaces: Problem 1
    • Isomorphic spaces: Problem 2
    • Isomorphic spaces: Problem 3
    • Vector spaces, fields, rings; ring homomorphisms and isomorphisms
    • Vector spaces, fields, rings, Problem 4
    • Vector spaces, fields, rings, Problem 5

    Recurrence relations, dynamical systems, Markov matrices

    • Continuous versus discrete
    • Two famous examples of recurrence
    • Linear discrete dynamical systems
    • Systems of difference equations, Problem 1
    • Systems of difference equations, Problem 2
    • Systems of difference equations, Problem 3
    • Higher order difference equations, Problem 4
    • Higher order difference equations, Problem 5
    • Higher order difference equations, Problem 6
    • Markov matrices
    • Each Markov matrix has eigenvalue 1
    • Steady-state vector (equilibrium vector), Problem 7
    • Markov matrices, Problem 8, Restaurant
    • Markov matrices, Problem 9, Migration
    • Markov matrices, Problem 10, Election
    • Dynamical systems, Problem 11
    • Where to read more on this topic?

    Solving systems of linear ODE, and solving higher order ODE

    • What is an ODE and what kinds of ODE we are going to deal with
    • Solutions to first order linear ODE with constant coefficients
    • Systems of first order linear ODE with constant coefficients
    • A very simple example
    • The method
    • System of ODE, Problem 1
    • System of ODE, Problem 2
    • System of ODE, Problem 3
    • How to deal with higher order linear ODE?
    • Another way of looking at the same problem

    Inner product as a generalization of dot product

    • Between concrete and abstract
    • Dot product in Part 1
    • Dot product and orthogonality in Part 2
    • From R^2 to inner product spaces
    • Inner product spaces
    • Euclidean n-space
    • A very important remark about notation
    • Inner and outer products
    • Weighted Euclidean inner product, Problem 1
    • Remember transposed matrices?
    • Positive definite matrices
    • Quadratic forms and how to read them
    • Matrix inner products on R^n, Problem 2
    • Gram matrix, Problem 3
    • Gram matrix, Problem 4
    • Inner product in the space of continuous functions
    • Gram matrix for an inner product in the space Pn of polynomials
    • Two inner products on the space of polynomials Pn
    • The evaluation inner products on P2, Problem 5
    • Inner product in the space of m × n matrices
    • Inner product in the space of square matrices
    • Inner product in the space of matrices, Problem 6
    • Frobenius inner product; Hadamard product of matrices

    Norm, distance, angles, and orthogonality in inner product spaces

    • Norm in inner product spaces
    • Weird geometry in the Euclidean space with weighted inner product
    • Frobenius norm of matrices, Problem 1
    • Norm in the space of functions, Problem 2
    • Distance in inner product spaces
    • Frobenius distance between matrices, Problem 3
    • Distance in the space of functions, Problem 4
    • First step to defining abstract angles
    • Cauchy–Schwarz inequality, proof 1
    • Cauchy–Schwarz inequality, proof 2
    • Cauchy–Schwarz inequality in the space of continuous functions
    • Angles in inner product spaces
    • More weird geometry: Angles in inner product spaces, Problem 5
    • Angles in inner product spaces, Problem 6
    • Orthogonality in inner product spaces
    • Orthogonality in inner product spaces depends on inner product
    • Orthogonality in inner product spaces, Problem 7
    • What is triangle inequality?
    • Triangle inequality in inner product spaces
    • Generalized Theorem of Pythagoras
    • Generalized Theorem of Pythagoras, Problem 8
    • Generalized Theorem of Pythagoras, Problem 9
    • Generalized Theorem of Pythagoras, Problem 10

    Projections and Gram–Schmidt process in various inner product spaces

    • Different but still awesome!
    • ON bases in IP spaces
    • Why does normalizing work in the same way in all IP spaces?
    • Orthonormal sets of continuous functions, Problem 1
    • Orthogonal complements, Problem 2
    • Orthogonal sets are linearly independent, Problem 3
    • Coordinates in orthogonal bases in IP spaces
    • Projections and orthogonal decomposition in IP spaces
    • Orthogonal projections on subspaces of an IP space, Problem 4
    • Orthogonal projections on subspaces of an IP space, Problem 5
    • Gram–Schmidt in IP spaces
    • Gram–Schmidt in IP spaces, Problem 6: Legendre polynomials
    • Gram–Schmidt in IP spaces, Problem 7
    • Easy computations of IP in ON bases, Problem 8

    Min-max problems, best approximations, and least squares

    • In this section
    • Min-max, Problem 1
    • Min-max, Problem 2
    • Min-max, Problem 3
    • Min-max, Problem 4
    • Min-max, Problem 5
    • Another look at orthogonal projections as matrix transformations
    • Orthogonal projections, Problem 6
    • Orthogonal projections, Problem 7
    • Shortest distance from a subspace
    • Shortest distance, Problem 8
    • Shortest distance, Problem 9
    • Shortest distance, Problem 10
    • Solvability of systems of equations in terms of the column space
    • Least squares solution and residual vector
    • Four fundamental matrix spaces and the normal equation
    • Least squares, Problem 11, by normal equation
    • Least squares, Problem 11, by projection
    • Least squares straight line fit, Problem 12
    • Least squares, fitting a quadratic curve to data, Problem 13

    Diagonalization of symmetric matrices

    • The link between symmetric matrices and quadratic forms, Problem 1
    • Some properties of symmetric matrices
    • Eigenvectors corresponding to distinct eigenvalues for a symmetric matrix
    • Complex numbers: a brief repetition
    • Eigenvalues for a (real) symmetric matrix are real
    • Orthogonal diagonalization
    • If a matrix is orthogonally diagonalizable, it is symmetric
    • The Spectral Theorem: Each symmetric matrix is orthogonally diagonalizable
    • Orthogonal diagonalization: how to do it
    • Orthogonal diagonalization, Problem 2
    • Spectral decomposition for symmetric matrices, Problem 3
    • Orthogonal diagonalization, Problem 4
    • Orthogonal diagonalization, Problem 5
    • Orthogonal diagonalization, Problem 6
    • Orthogonal diagonalization, Problem 7
    • Spectral decomposition, Problem 8
    • Pos and neg definite matrices, semidefinite and indefinite matrices, Problem 9
    • The wonderful strength of an orthogonally diagonalized matrix
    • Three tests for definiteness of symmetric matrices, Problem 10
    • Symmetric square roots of symmetric positive definite matrices; singular values

    Quadratic forms and their classification

    • The correspondence between quadratic forms and symmetric matrices is 1-to-1
    • Completing the square is not unique
    • What kind of questions we want to answer
    • 163 Quadratic forms in two variables, Problem 1.
    • Quadratic forms in two variables, Problem 2
    • Quadratic curves, generally
    • Quadratic curves as conic sections
    • Quadratic curves by distances; shortest distance from the origin
    • Principal axes; The shortest distance from the origin, Problem 3
    • Classification of quadratic forms in two variables
    • Classification of curves, Problem 4
    • Classification of curves, Problem 5
    • Different roles of symmetric matrices (back to Videos 150 and 168), Problem
    • Classification of curves, Problem 7
    • Generally about quadratic surfaces
    • Some nice visuals on quadratic surfaces
    • Quadratic surfaces, shortest distance, Problem 8
    • Quadratic surfaces, Problem 9
    • Quadratic surfaces, Problem 10
    • Law of inertia for quadratic forms; Signature of a form, Problem 11
    • Four methods of determining definiteness; Problem 12

    Constrained optimization

    • The theory for this section
    • Constrained optimization, Problem 1
    • Constrained optimization, Problem 2
    • Constrained optimization, Problem 3
    • Constrained optimization, Problem 4

    The Grand Finale: Singular Value Decomposition and Pseudoinverses

    • All our roads led us to SVD
    • Why do we need SVD?
    • We know really a lot about AT A for any rectangular matrix A
    • New facts about AT A: eigenvalues and eigenvectors Singular values of A
    • ON-bases containing only eigenvectors of certain matrix products
    • Singular value decomposition with proof and geometric interpretation
    • SVD, reduced singular value decomposition, Problem 1
    • SVD, Problem 2
    • More new facts about AT A: six equivalent statements
    • Least squares, SVD, and pseudoinverse (Moore–Penrose inverse)
    • Pseudoinverse, Problem 3
    • SVD and Fundamental Theorem of Linear Algebra

    Wrap-up Linear Algebra and Geometry

    • Linear Algebra and Geometry, Wrap-up
    • So, what’s next?
    • Final words

    Extras

    • Bonus Lecture

    Instructors

    Ms Hania Uscka Wehlou

    Ms Hania Uscka Wehlou
    Teacher in mathematics
    Udemy

    Ph.D

    Mr Martin Wehlou

    Mr Martin Wehlou
    Editor
    Freelancer

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