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    Linear Algebra and Geometry 2

    BY
    Udemy

    Acquire a thorough understanding of the fundamental concepts of geometry 2 and linear algebra, as well as knowledge of numerous matrices and vectors.

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    Mode

    Online

    Fees

    ₹ 3299

    Quick Facts

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

    Course overview

    Geometry 2 and linear algebra are mathematical disciplines that are largely concerned with algebra's inner product spaces, such as real and complex vector spaces, projections, and orthogonal and orthonormal spaces. This also covers the various types of matrices, such as linear and orthogonal matrices. Hania Uscka-Wehlou, Ph.D. and Mathematics Teacher, collaborated with Martin Wehlou, Certified Editor at MITM, to produce the Linear Algebra and Geometry 2 online certification, which is available through Udemy.

    Linear Algebra and Geometry 2 online course is a self-paced training program that focuses on assisting learners in mastering the fundamentals and advanced principles of geometry and linear algebra. Learners in the Linear Algebra and Geometry 2 online classes will have access to 47 hours of video-based learning materials, as well as 331 downloadable resources, covering topics such as systems of equations, vectors, vector spaces, matrices, linear dependencies, linear independencies, linear transformation, geometrical transformation, eigenspaces, eigenvalues, eigenvectors, and the Gram-Schmidt process, among others.

    The highlights

    • Certificate of completion
    • Self-paced course
    • 47 hours of pre-recorded video content
    • 331 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
    ₹ 3,299
    certificate availability

    Yes

    certificate providing authority

    Udemy

    Who it is for

    Teacher Scientist

    What you will learn

    Mathematical skill

    After completing the Linear Algebra and Geometry 2 certification course, learners will gain an insight into the fundamentals of linear algebra and geometry 2. Learners will explore matrices, vectors, vector spaces, isometry, eigenvalues, eigenvectors, eigenspaces, vandermonde determinants, and the system of equations, among other mathematical concepts utilized in both geometry and algebra. Learners will study transition matrices, orthogonal matrices, linear dependencies, linear independencies, linear combinations, linear transformations, geometrical transformations, and other methodologies.

    The syllabus

    Introduction to the course

    • Introduction to the course

    Real vector spaces and their subspaces

    • From abstract to concrete
    • From concrete to abstract
    • Our prototype
    • Formal definition of vector spaces Example 1: Rn
    • Vector spaces, Example 2: m × n matrices with real entries
    • Vector spaces, Example 3: real-valued functions on some interval
    • Vector spaces, Example 4: complex numbers
    • Cancellation property
    • Two properties of vector spaces; Definition of difference
    • Some properties of vector spaces
    • What is a subspace
    • All the subspaces in R2
    • All the subspaces in R3
    • Subspaces, Problem 1
    • Subspaces, Problem 2
    • Subspaces, Problem 3
    • Subspaces, Problem 4

    Linear combinations and linear independence

    • Our unifying example
    • Linear combinations in Part 1
    • Linear combinations, new stuff. Example 1
    • Linear combinations Example 2
    • Linear combinations, Problem 1
    • Linear combinations, Problem 2
    • What is a span, definition and some examples
    • Span, Problem 3
    • Span, Problem 4
    • Span, Problem 5
    • What do we mean by trivial?
    • Linear independence and linear dependence
    • Geometry of linear independence and linear dependence
    • An important remark on linear independence in Rn
    • Linearly independent generators, Problem 6
    • Linear independence in the set of matrices, Problem 7
    • Linear independence in C^0[−∞, ∞], Problem 8
    • Vandermonde determinant and polynomials
    • Linear independence in C^∞(R), Problem 9
    • Wronskian and linear independence in C∞(R)
    • Linear independence in C^∞(R), Problem 10
    • Linear independence in C^∞(R), Problem 11

    Coordinates, basis, and dimension

    • What is a basis and dimension?
    • Bases in the 3-space, Problem 1
    • Bases in the plane and in the 3-space
    • Bases in the 3-space, Problem 2
    • Bases in the 4-space, Problem 3
    • Bases in the 4-space, Problem 4
    • Bases in the space of polynomials, Problem 5
    • Coordinates with respect to a basis
    • Coordinates with respect to a basis are unique
    • Coordinates in our unifying example
    • Dimension of a subspace, Problem 6
    • Bases in a space of functions, Problem 7

    Change of basis

    • Coordinates in different bases
    • It is easy to recalculate from the standard basis
    • Transition matrix, a derivation
    • Previous example with transition matrix
    • Our unifying example
    • One more simple example and bases
    • Two non-standard bases, Method 1
    • Two non-standard bases, Method 2
    • How to recalculate coordinates between two non-standard bases? An algorithm
    • Change of basis, Problem 1
    • Change of basis, Problem 2
    • Change of basis, Problem 3
    • Change of basis, Problem 4
    • Change of basis, Problem 5
    • Change to an orthonormal basis in R^2

    Row space, column space, and nullspace of a matrix

    • What you are going to learn in this section
    • Row space and column space for a matrix
    • What are the elementary row operations doing to the row spaces?
    • What are the elementary row operations doing to the column spaces?
    • Column space, Problem 2
    • Determining a basis for a span, Problem 3
    • Determining a basis for a span consisting of a subset of given vectors, Prob
    • Determining a basis for a span consisting of a subset of given vectors, Prob
    • A tricky one: Let rows become columns, Problem 6
    • A basis in the space of polynomials, Problem 7
    • Nullspace for a matrix
    • How to find the nullspace, Problem 8
    • Nullspace, Problem 9
    • Nullspace, Problem 10

    Rank, nullity, and four fundamental matrix spaces

    • Rank of a matrix
    • Nullity
    • Relationship between rank and nullity
    • Relationship between rank and nullity, Problem 1
    • Relationship between rank and nullity, Problem 2
    • Relationship between rank and nullity, Problem 3
    • Orthogonal complements, Problem 4
    • Four fundamental matrix spaces
    • The Fundamental Theorem of Linear Algebra and Gilbert Strang

    Matrix transformations from R^n to R^m

    • What do we mean by linear?
    • Some terminology
    • How to think about functions from Rn to Rm?
    • When is a function from Rn to Rm linear? Approach 1
    • When is a function from Rn to Rm linear? Approach 2
    • When is a function from Rn to Rm linear? Approach 3
    • Approaches 2 and 3 are equivalent
    • Matrix transformations, Problem 1
    • Image, kernel, and inverse operators, Problem 2
    • Basis for the image, Problem 3
    • Kernel, Problem 4
    • Image and kernel, Problem 5
    • Inverse operators, Problem 6
    • Linear transformations, Problem 7
    • Kernel and geometry, Problem 8
    • Linear transformations, Problem 9

    Geometry of matrix transformations on R^2 and R^3

    • Our unifying example: linear transformations and change of basis
    • An example with nontrivial kernel
    • Line symmetries in the plane
    • Projection on a given vector, Problem 1
    • Symmetry about the line y = kx, Problem 2
    • Rotation by 90 degrees about the origin
    • Rotation by the angle α about the origin
    • Expansion, compression, scaling, and shear
    • Plane symmetry in the 3-space, Problem 3
    • Projections on planes in the 3-space, Problem 4
    • Symmetry about a given plane, Problem 5
    • Projection on a given plane, Problem 6
    • Rotations in the 3-space, Problem 7

    Properties of matrix transformations

    • What kind of properties we will discuss
    • What happens with vector subspaces and affine subspaces under linear transfo
    • Parallel lines transform into parallel lines, Problem 1
    • Transformations of straight lines, Problem 2
    • Change of area (volume) under linear operators in the plane (space)
    • Change of area under linear transformations, Problem 3
    • Compositions of linear transformations
    • How to obtain the standard matrix of a composition of linear transformations
    • Why does it work?
    • Compositions of linear transformations, Problem 4
    • Compositions of linear transformations, Problem 5

    General linear transformations in different bases

    • Linear transformations between two linear spaces
    • Linear transformations, Problem 1
    • Linear transformations, Problem 2
    • Linear transformations, Problem 3
    • Linear transformations, Problem 4
    • Linear transformations, Problem 5
    • Linear transformations in different bases, Problem 6
    • Linear transformations in different bases
    • Linear transformations in different bases, Problem 7
    • Linear transformations in different bases, Problem 8
    • Linear transformations in different bases, Problem 9
    • Linear transformations, Problem 10
    • Linear transformations, Problem 11

    Gram–Schmidt process

    • Dot product and orthogonality until now
    • Orthonormal bases are awesome
    • Orthonormal bases are awesome, Reason 1: distance
    • Orthonormal bases are awesome, Reason 2: dot product
    • Orthonormal bases are awesome, Reason 3: transition matrix
    • Orthonormal bases are awesome, Reason 4: coordinates
    • Coordinates in ON bases, Problem 1
    • Coordinates in orthogonal bases, Theorem and proof
    • Each orthogonal set is linearly independent, Proof
    • Coordinates in orthogonal bases, Problem 2
    • Orthonormal bases, Problem 3
    • Projection Theorem 1
    • Projection Theorem 2
    • Projection Formula, an illustration in the 3-space
    • Calculating projections, Problem 4
    • Calculating projections, Problem 5
    • Gram–Schmidt Process
    • Gram–Schmidt Process, Our unifying example
    • Gram–Schmidt Process, Problem 6
    • Gram–Schmidt Process, Problem 7

    Orthogonal matrices

    • Product of a matrix and its transposed is symmetric
    • Definition and examples of orthogonal matrices
    • Geometry of 2-by-2 orthogonal matrices
    • A 3-by-3 example
    • Useful formulas for the coming proofs
    • Property 1: Determinant of each orthogonal matrix is 1 or −1
    • Property 2: Each orthogonal matrix A is invertible and A−1 is also orthogona
    • Property 3: Orthonormal columns and rows
    • Property 4: Orthogonal matrices are transition matrices between ON-bases
    • Property 5: Preserving distances and angles
    • Property 6: Product of orthogonal matrices is orthogonal
    • Orthogonal matrices, Problem 1
    • Orthogonal matrices, Problem 2

    Eigenvalues and eigenvectors

    • Crash course in factoring polynomials
    • Eigenvalues and eigenvectors, the terms
    • Order of defining, order of computing
    • Eigenvalues and eigenvectors geometrically
    • Eigenvalues and eigenvectors, Problem 1
    • How to compute eigenvalues Characteristic polynomial
    • How to compute eigenvectors
    • Finding eigenvalues and eigenvectors: short and sweet
    • Eigenvalues and eigenvectors for examples from Video 180
    • Eigenvalues and eigenvectors, Problem 3
    • Eigenvalues and eigenvectors, Problem 4
    • Eigenvalues and eigenvectors, Problem 5
    • Eigenvalues and eigenvectors, Problem 6
    • Eigenvalues and eigenvectors, Problem 7

    Diagonalization

    • Why you should love diagonal matrices
    • Similar matrices
    • Similarity of matrices is an equivalence relation (RST)
    • Shared properties of similar matrices
    • Diagonalizable matrices
    • How to diagonalize a matrix, a recipe
    • Diagonalize our favourite matrix
    • Eigenspaces; geometric and algebraic multiplicity of eigenvalues
    • Eigenspaces, Problem 2
    • Eigenvectors corresponding to different eigenvalues are linearly independent
    • A sufficient, but not necessary, condition for diagonalizability
    • Necessary and sufficient condition for diagonalizability
    • Diagonalizability, Problem 3
    • Diagonalizability, Problem 4
    • Diagonalizability, Problem 5
    • Diagonalizability, Problem 6
    • Diagonalizability, Problem 7
    • Powers of matrices
    • Powers of matrices, Problem 8
    • Diagonalization, Problem 9
    • Sneak peek into the next course; orthogonal diagonalization

    Wrap-up Linear Algebra and Geometry 2

    • Linear Algebra and Geometry 2, Wrap-up
    • Yes, there will be Part 3!
    • 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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