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    Linear Algebra and Geometry 1

    BY
    Udemy

    Acquire a hands-on understanding of the foundational strategies associated with linear algebra and geometry.

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    Mode

    Online

    Fees

    ₹ 2999

    Quick Facts

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

    Course overview

    Algebraic geometry is a field of mathematics that examines multidimensional polynomial zeros. For answering geometrical questions regarding these sets of zeros, contemporary algebraic geometry relies on theoretical algebraic techniques, namely commutative algebra. Linear Algebra and Geometry 1 online certification has been created by Hania Uscka-Wehlou, University Professor in Mathematics in association with Martin Wehlou - Editor at MITM AB which is offered by Udemy.

    Linear Algebra and Geometry 1 online course is to educate students on how to solve problems in linear algebra and geometry, as well as why these methods are used so frequently. Linear Algebra and Geometry 1 online training include 46 hours of video-based learning material and 358 downloadable study materials covering topics such as matrix, matrix inverse, vector, linear equations, Jacobi algorithm matrix multiplication, vector decomposition, orthogonal projection, the system of equations, back substitution, Gaussian elimination, and Gauss-Jordon elimination.

    The highlights

    • Certificate of completion
    • Self-paced course
    • 46 hours of pre-recorded video content
    • 358 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
    ₹ 2,999
    certificate availability

    Yes

    certificate providing authority

    Udemy

    Who it is for

    Scientist

    What you will learn

    Problem solving ability Mathematical skill

    After completing the Linear Algebra and Geometry 1 certification course, students will develop a deeper knowledge of mathematical concepts like geometry and linear algebra. Students will learn how to solve linear equations, determinant equations, and systems of equations using a range of methodologies. Students will study vectors, matrices, inverse matrices, and the Jacobi algorithm, as well as theories such as invertible matrix theory, Gaussian elimination, and Gauss-Jordan elimination. Students will also gain a thorough understanding of matric multiplication, vector decomposition, and orthogonal projection, as well as problem-solving skills.

    The syllabus

    Introduction to the course

    • Introduction

    Some basic concepts

    • Coordinate systems and coordinates in the plane and in the 3-space.
    • Slope-intercept equations of straight lines in the plane
    • Normal equations of planes in the 3-space.
    • Vectors
    • Scalars
    • Vector addition and vector scaling
    • Linear combinations
    • Matrices
    • Linear transformations
    • Matrix—vector multiplication
    • Rules for computations with real numbers
    • Pythagorean Theorem and distance between points
    • Sine, cosine, and pythagorean identity
    • Cosine Rule

    Systems of linear equations; building up your geometrical intuition

    • Different ways of looking at equations
    • Solution set
    • Linear and non-linear equations
    • Systems of linear equations
    • Solution sets of systems of linear equations
    • An example of a 2 × 2 system of linear equations, a graphical solution
    • Possible solution sets of 2 × 2 systems of linear equations
    • Possible solution sets of 3 × 2 systems of linear equations
    • Possible solution sets of 3 × 3 systems of linear equations
    • Possible solution sets of 2 × 3 systems of linear equations
    • Possible solution sets of m × n systems of linear equations

    Solving systems of linear equations; Gaussian elimination

    • Our earlier problem revisited; an algebraical solution
    • Three elementary operations
    • What is Gauss—Jordan elimination and Gaussian elimination?
    • Gauss—Jordan elimination, a 2-by-2 system with unique solution
    • The same example solved with Gaussian elimination and back-substitution
    • The same example solved with matrix operations; coefficient matrix and augmented
    • How to write the augmented matrix for a given system of equations, Problem 1
    • How to write system of equations to a given augmented matrix, Problem 2
    • Gaussian elimination, Problem 3
    • Gaussian elimination, Problem 4
    • Gaussian elimination, Problem 5
    • Gaussian elimination, Problem 6.
    • What happens if the system is inconsistent?
    • Gaussian elimination, Problem 7.
    • Preparation to the general formulation of the algorithm; REF and RREF matrices.
    • How to read solutions from REF and RREF matrices?
    • General formulation of the algorithm in Gauss–Jordan elimination
    • Gauss–Jordan elimination, Problem 8
    • Gauss–Jordan elimination, Problem 9
    • Gaussian elimination, Problem 10
    • Gauss–Jordan elimination, Problem 11
    • Gauss–Jordan elimination, Problem 12
    • Gauss–Jordan elimination, Problem 13

    Some applications in mathematics and natural sciences

    • Solving systems of linear equations in Linear Algebra and Geometry
    • Solving systems of linear equations (Calculus) Problem 1
    • Solving systems of linear equations (Calculus) Problem 2
    • Solving systems of linear equations (Calculus) Problem 3
    • Solving systems of linear equations (Calculus) Problem 4
    • Problem 5 (Chemistry)
    • Problem 6 (Electrical circuits)

    Matrices and matrix operations

    • Introduction to matrices
    • Different types of matrices
    • Matrix addition and subtraction, Problem 1
    • Matrix scaling, with geometrical interpretation
    • Matrix scaling, Problem 2
    • Matrix multiplication, with geometrical interpretation
    • Matrix multiplication, how to do
    • Matrix multiplication, Problem 3
    • Matrix multiplication and systems of equations, Problem 4
    • Transposed matrix, definition and some examples
    • Trace of a matrix, definition and an example
    • Various matrix operations, Problem 7
    • Various matrix operations, Problem 8

    Inverses; Algebraic properties of matrices

    • Properties of matrix operations, an introduction
    • Matrix addition has all the good properties
    • Matrix multiplication has a neutral element for square matrices
    • Matrix multiplication is associative
    • Matrix multiplication is not commutative
    • Sometimes commutativity happens, Problem 1
    • Two distributive laws
    • Matrix multiplication does not have the zero-product property
    • There is no cancellation law for matrix multiplication
    • Inverse matrices; not all non-zero square matrices have an inverse
    • Inverse matrix for 2-by-2 matrices; non-zero determinant
    • Solving matrix equations, Problem 2
    • Powers of matrices; powers of diagonal matrices
    • Computation rules for transposed matrices
    • Supplement to Video 83; Inverse of a product
    • Inverse of a transposed matrix
    • Various rules, Problem 3

    Elementary matrices and a method for finding A inverse

    • Inverse matrices, introduction to the algorithm
    • Algorithm for inverse matrices, an example
    • Matrix inverse, Problem 1
    • Matrix inverse, Problem 2
    • Matrix equations, Problem 3
    • Matrix equations, Problem 4
    • Matrix equations, Problem 5
    • Matrix equations, Problem 6
    • Matrix inverse, Problem 7
    • Elementary operations and elementary matrices
    • Inverse elementary operations and their matrices
    • A really important theorem
    • Four equivalent statements

    Linear systems and matrices

    • Formally about the number of solutions to systems of linear equations
    • Two more statements in our important theorem
    • Solution of a linear system using A inverse, Problem 1
    • Determining consistency by elimination, Problem 2
    • Matrix equations, Problem 3

    Determinants

    • Why the determinants are important
    • 2-by-2 determinants; notation for n-by-n determinants
    • Geometrical interpretations of determinants
    • Geometrically about the determinant of a product
    • Definition of determinants
    • Conclusion 1: Determinant of matrices with interchanged columns
    • Conclusion 2: What happens when one column is a linear combination of others
    • Conclusion 3: About adding a multiple of a column to another column
    • Conclusion 4: Determinant of kA for any k ∈ R
    • Elementary column operations
    • How to compute 2-by-2 determinants from the definition
    • How to compute 3-by-3 determinants from the definition
    • Sarrus’ rule for 3-by-3 determinants
    • Determinant of transposed matrix; row operations
    • Evaluating determinants by cofactor expansion along rows or columns
    • Evaluating determinants by row or column reduction
    • Determinant of inverse
    • Properties of determinants, Problem 1
    • Properties of determinants, Problem 2
    • Properties of determinants, Problem 3
    • Determinant equations, Problem 4
    • Determinant equations, Problem 5
    • Determinant equations, Problem 6
    • Determinant equations, Problem 7
    • Invertible matrices, determinant test with a proof, Problem 8
    • Cramer’s rule, a proof, an example, and a geometrical interpretation
    • Cramer’s rule, Problem 9
    • Inverse matrix, an explicit formula
    • Invertible matrices, Problem 10
    • Problem 11, a large determinant
    • Problem 12, another large determinant
    • Problem 13: a trigonometric determinant
    • Problem 14: Vandermonde determinant

    Vectors in 2-space, 3-space, and n-space

    • Vectors, a repetition
    • Computation rules for vector addition and scaling
    • Computations with vectors, Problem 1
    • Computations with vectors, Problem 2
    • Computations with vectors, Problem 3
    • Parallel vectors, Problem 4
    • Parallel vectors, Problem 5
    • Linear combinations, Problem 6
    • Linear combinations, Problem 7
    • Linear combinations, linear independence, Problem 8
    • Linear combinations, linear dependence, Problem 9
    • Area, Problem 10
    • Midpoint of a line segment, Problem 11

    Distance and norm in R^n

    • Norm of a vector, Problem 1
    • Properties of the norm
    • Distance between points, Problem 2
    • Unit vectors, how to normalize a vector
    • Unit vectors in given direction, Problem 3

    Dot product, orthogonality, and orthogonal projections

    • Different products for vectors
    • Perpendicular straight lines and orthogonal vectors
    • Orthogonal projections
    • Definition of dot product for geometrical vectors
    • How to compute dot product, an example
    • Dot product for vectors in R^n; orthogonality and angles
    • Properties of dot product
    • Angles between vectors, Problem 1
    • Angles between vectors, Problem 2
    • How to find vector orthogonal to a given vector in the plane or in the 3-space
    • Orthogonal projections and decompositions
    • Orthogonal projections and decompositions, Problem 3
    • Orthogonal projections and decompositions, Problem 4
    • Orthogonal sets, Problem 5
    • Orthogonal projections and decompositions, Problem 6

    Cross product, parallelograms and parallelepipeds

    • Cross product, an introduction
    • Cross product, how it is defined
    • Three properties of cross product
    • The length of the cross product of two vectors
    • More properties of cross product
    • Cross product: Problem 1
    • Cross product in the plane
    • Scalar triple product and volume
    • Scalar triple product, Problem 2
    • Collinearity in the plane and coplanarity in the 3-space
    • Determinant test for vectors, Problem 3

    Lines in R^2

    • Lines in the plane, an introduction
    • Slope-intercept and intercept form
    • Normal equation
    • Parametric equations
    • Determinant equation
    • Lines in the plane, Problem 5

    Planes in R^3

    • Planes in the 3-space, an introduction
    • Normal and intercept equation
    • Parametric equations
    • Parametric to normal
    • Normal to parametric; find a point and two parallel vectors for a given plane
    • Determinant equation
    • Planes: Problem 6

    Lines in R^3

    • Lines in the 3 space, an introduction
    • Lines in the 3 space, Problem 1
    • Lines in the 3 space, Problem 2
    • Lines in the 3 space, Problem 3

    Geometry of linear systems; incidence between lines and planes

    • Incidence 1: points and planes
    • Incidence 2: planes and lines
    • Incidence 3: points and lines
    • Parallel and orthogonal objects
    • Parallel planes, Problem 4
    • Parallel planes: Problem 5
    • Orthogonal planes: Problem 6
    • Planes and lines, Problem 7
    • Planes and lines, Problem 8
    • Planes, Problem 9
    • Planes, lines, and systems of equations, Problem 10

    Distance between points, lines, and planes

    • Distances between sets, generally
    • Distance between points and planes
    • Distance between points and planes, Problem 1
    • Distance between points and planes, Problem 2
    • Distance between points and lines
    • Distance between points and lines, Problem 3
    • Distance between points and lines, Problem 4
    • Distance between (skew) lines
    • Distance between (skew) lines, Problem 5
    • Distance between (skew) lines, Problem 6

    Some words about the next course

    • Linear Algebra and Geometry 1, Wrap-up
    • Linear Algebra and Geometry 2, some words about it
    • Final words

    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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