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    Complete linear algebra: theory and implementation in code

    BY
    Udemy

    Learn how to utilize the functionalities of MATLAB and Python to apply linear algebra and matrix analysis concepts.

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    Mode

    Online

    Fees

    ₹ 3499

    Quick Facts

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

    Course overview

    One of the most significant areas of mathematics is linear algebra. The subject of vectors and linear equations is known as linear algebra. It is a fundamental notion in practically every field of mathematics. In the current teaching of geometry, linear algebra is considered a fundamental concept. Mike X Cohen - Neuroscientist, Writer, and Professor created the Complete Linear Algebra: Theory and Implementation in a code certification course, which is delivered via Udemy.

    Complete linear algebra: theory and implementation in code online classes are designed for candidates who work in fields like data science, artificial intelligence, computer science, and machine learning as well as those who want to utilize the functionalities of the principal and strategies associated with linear algebra. Complete linear algebra: theory and implementation in code online course consist of 34 hours of detailed video sessions accompanied by 2 articles and 3 downloadable resources that cover a wide range of linear algebra topics such as quadratic form, the system of equations, orthogonalization, singular value decomposition, eigendecomposition, and more.

    The highlights

    • Certificate of completion
    • Self-paced course
    • 34 hours of pre-recorded video content
    • 2 articles
    • 3 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
    ₹ 3,499
    certificate availability

    Yes

    certificate providing authority

    Udemy

    What you will learn

    Knowledge of python Mathematical skill Machine learning Knowledge of artificial intelligence

    After completing the Complete linear algebra: theory and implementation in code online certification, candidates will gain an understanding of the concepts and principles involved with mathematics and linear algebra. Candidates will learn how to use linear algebra in various applications such as Matlab, Python, machine learning, and artificial intelligence. Candidates will study vectors, matrices, matrix multiplication, matrix rank, matrix spaces, matrix determinants, matrix inverse, least squares, and quadratic forms, among other topics in linear algebra. Candidates will also acquire knowledge of the concepts like projections, orthogonalization, eigendecomposition, and singular value decomposition.

    The syllabus

    Introductions

    • What is linear algebra?
    • Linear algebra applications
    • An enticing start to a linear algebra course!
    • How best to learn from this course
    • Maximizing your Udemy experience

    Get the course materials

    • How to download and use course materials

    Vectors

    • Exercises + code
    • Algebraic and geometric interpretations of vectors
    • Vector addition and subtraction
    • Vector-scalar multiplication
    • Vector-vector multiplication: the dot product
    • Dot product properties: associative, distributive, commutative
    • Code challenge: dot products with matrix columns
    • Code challenge: is the dot product commutative?
    • Vector length
    • Vector length in MATLAB
    • Vector length in Python
    • Dot product geometry: sign and orthogonality
    • Vector orthogonality
    • Code challenge: Cauchy-Schwarz inequality
    • Relative vector angles
    • Code challenge: dot product sign and scalar multiplication
    • Vector Hadamard multiplication
    • Outer product
    • Vector cross product
    • Vectors with complex numbers
    • Hermitian transpose (a.k.a. conjugate transpose)
    • Interpreting and creating unit vectors
    • Code challenge: dot products with unit vectors
    • Dimensions and fields in linear algebra
    • Subspaces
    • Subspaces vs. subsets
    • Span
    • In the span?
    • Linear independence
    • Basis

    Introduction to matrices

    • Matrix terminology and dimensionality
    • Matrix sizes and dimensionality
    • A zoo of matrices
    • Can the matrices be concatenated?
    • Matrix addition and subtraction
    • Matrix-scalar multiplication
    • Code challenge: is matrix-scalar multiplication a linear operation?
    • Transpose
    • Complex matrices
    • Addition, equality, and transpose
    • Diagonal and trace
    • Code challenge: linearity of trace
    • Broadcasting matrix arithmetic

    Matrix multiplications

    • Introduction to standard matrix multiplication
    • Four ways to think about matrix multiplication
    • Code challenge: matrix multiplication by layering
    • Matrix multiplication with a diagonal matrix
    • Order-of-operations on matrices
    • Matrix-vector multiplication
    • Find the missing value!
    • 2D transformation matrices
    • Code challenge: Pure and impure rotation matrices
    • Code challenge: Geometric transformations via matrix multiplications
    • Additive and multiplicative matrix identities
    • Additive and multiplicative symmetric matrices
    • Hadamard (element-wise) multiplication
    • Matrix operation equality
    • Code challenge: symmetry of combined symmetric matrices
    • Multiplication of two symmetric matrices
    • Code challenge: standard and Hadamard multiplication for diagonal matrices
    • Code challenge: Fourier transform via matrix multiplication!
    • Frobenius dot product
    • Matrix norms
    • Code challenge: conditions for self-adjoint
    • Code challenge: The matrix asymmetry index
    • What about matrix division?

    Matrix rank

    • Rank: concepts, terms, and applications
    • Maximum possible rank.
    • Computing rank: theory and practice
    • Rank of added and multiplied matrices
    • What's the maximum possible rank?
    • Code challenge: reduced-rank matrix via multiplication
    • Code challenge: scalar multiplication and rank
    • Rank of A^TA and AA^T
    • Code challenge: rank of multiplied and summed matrices
    • Making a matrix full-rank by "shifting"
    • Code challenge: is this vector in the span of this set?
    • Course tangent: self-accountability in online learning

    Matrix spaces

    • Column space of a matrix
    • Column space, visualized in code
    • Row space of a matrix
    • Null space and left null space of a matrix
    • Column/left-null and row/null spaces are orthogonal
    • Dimensions of column/row/null spaces
    • Example of the four subspaces
    • More on Ax=b and Ax=0

    Solving systems of equations

    • Systems of equations: algebra and geometry
    • Converting systems of equations to matrix equations
    • Gaussian elimination
    • Echelon form and pivots
    • Reduced row echelon form
    • Code challenge: RREF of matrices with different sizes and ranks
    • Matrix spaces after row reduction

    Matrix determinant

    • Determinant: concept and applications
    • Determinant of a 2x2 matrix
    • Code challenge: determinant of small and large singular matrices
    • Determinant of a 3x3 matrix
    • Code challenge: large matrices with row exchanges
    • Find matrix values for a given determinant
    • Code challenge: determinant of shifted matrices
    • Code challenge: determinant of matrix product

    Matrix inverse

    • Matrix inverse: Concept and applications
    • Computing the inverse in code
    • Inverse of a 2x2 matrix
    • The MCA algorithm to compute the inverse
    • Code challenge: Implement the MCA algorithm!!
    • Computing the inverse via row reduction
    • Code challenge: inverse of a diagonal matrix
    • Left inverse and right inverse
    • One-sided inverses in code
    • Proof: the inverse is unique
    • Pseudo-inverse, part 1
    • Code challenge: pseudoinverse of invertible matrices
    • Why should you avoid the inverse?

    Projections and orthogonalization

    • Projections in R^2
    • Projections in R^N
    • Orthogonal and parallel vector components
    • Code challenge: decompose vector to orthogonal components
    • Orthogonal matrices
    • Gram-Schmidt procedure
    • QR decomposition
    • Code challenge: Gram-Schmidt algorithm
    • Matrix inverse via QR decomposition
    • Code challenge: Inverse via QR
    • Code challenge: Prove and demonstrate the Sherman-Morrison inverse
    • Code challenge: A^TA = R^TR

    Least-squares for model-fitting in statistics

    • Introduction to least-squares
    • Least-squares via left inverse
    • Least-squares via orthogonal projection
    • Least-squares via row-reduction
    • Model-predicted values and residuals
    • Least-squares application 1
    • Least-squares application 2
    • Code challenge: Least squares via QR decomposition

    Eigendecomposition

    • What are eigenvalues and eigenvectors?
    • Finding eigenvalues
    • Shortcut for eigenvalues of a 2x2 matrix
    • Code challenge: eigenvalues of diagonal and triangular matrices
    • Code challenge: eigenvalues of random matrices
    • Finding eigenvectors
    • Eigendecomposition by hand: two examples
    • Diagonalization
    • Matrix powers via diagonalization
    • Code challenge: eigendecomposition of matrix differences
    • Eigenvectors of distinct eigenvalues
    • Eigenvectors of repeated eigenvalues
    • Eigendecomposition of symmetric matrices
    • Eigenlayers of a matrix
    • Code challenge: reconstruct a matrix from eigenlayers
    • Eigendecomposition of singular matrices
    • Code challenge: trace and determinant, eigenvalues sum and product
    • Generalized eigendecomposition
    • Code challenge: GED in small and large matrices

    Singular value decomposition

    • Singular value decomposition (SVD)
    • Are these two expressions equal?
    • Code challenge: SVD vs. eigendecomposition for square symmetric matrices
    • Relation between singular values and eigenvalues
    • Code challenge: U from eigendecomposition of A^TA
    • Code challenge: A^TA, Av, and singular vectors
    • SVD and the four subspaces
    • Spectral theory of matrices
    • SVD for low-rank approximations
    • Convert singular values to percent variance
    • Code challenge: When is UV^T valid, what is its norm, and is it orthogonal?
    • Singular values of an orthogonal matrix
    • SVD, matrix inverse, and pseudoinverse
    • SVD, (pseudo)inverse, and left-inverse
    • Condition number of a matrix
    • Code challenge: Create a matrix with desired condition number
    • Code challenge: Why do you avoid the inverse

    Quadratic form and definiteness

    • The quadratic form in algebra
    • The quadratic form in geometry
    • The normalized quadratic form
    • Code challenge: Visualize the normalized quadratic form
    • Eigenvectors and the quadratic form surface
    • Application of the normalized quadratic form: PCA
    • Quadratic form of generalized eigendecomposition
    • Matrix definiteness, geometry, and eigenvalues
    • Proof: A^TA is always positive (semi)definite
    • Proof: Eigenvalues and matrix definiteness

    Bonus section

    • Bonus lecture

    Instructors

    Mr Mike X Cohen

    Mr Mike X Cohen
    Associate Professor
    Freelancer

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