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Премиум
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    What we'll learn in this course
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    How to get the most out of this course
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    Introduction to operations on one matrix
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    Linear systems in two unknowns
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    Linear systems in three unknowns
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    Matrix dimensions and entries
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    Representing systems with matrices
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    Simple row operations
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    Pivot entries and row-echelon forms
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    Gauss-Jordan elimination
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    Number of solutions to the linear system
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    Introduction to operations on two matrices
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    Matrix addition and subtraction
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    Scalar multiplication
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    Zero matrices
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    Matrix multiplication
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    Identity matrices
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    The elimination matrix
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    Introduction to matrices as vectors
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    Vectors
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    Vector operations
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    Unit vectors and basis vectors
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    Linear combinations and span
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    Linear independence in two dimensions
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    Linear independence in three dimensions
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    Linear subspaces
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    Spans as subspaces
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    Basis
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    Introduction to dot products and cross products
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    Dot products
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    Cauchy-Schwarz inequality
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    Vector triangle inequality
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    Angle between vectors
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    Equation of a plane, and normal vectors
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    Cross products
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    Dot and cross products as opposite ideas
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    Introduction to matrix-vector products
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    Multiplying matrices by vectors
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    The null space and Ax=O
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    Null space of a matrix
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    The column space and Ax=b
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    Solving Ax=b
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    Dimensionality, nullity, and rank
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    Introduction to transformations
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    Functions and transformations
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    Transformation matrices and the image of the subset
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    Preimage, image, and the kernel
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    Linear transformations as matrix-vector products
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    Linear transformations as rotations
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    Adding and scaling linear transformations
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    Projections as linear transformations
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    Compositions of linear transformations
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    Introduction to inverses
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    Inverse of a transformation
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    Invertibility from the matrix-vector product
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    Inverse transformations are linear
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    Matrix inverses, and invertible and singular matrices
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    Solving systems with inverse matrices
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    Introduction to determinants
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    Determinants
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    Cramer's rule for solving systems
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    Modifying determinants
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    Upper and lower triangular matrices
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    Using determinants to find area
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    Introduction to transposes
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    Transposes and their determinants
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    Transposes of products, sums, and inverses
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    Null and column spaces of the transpose
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    The product of a matrix and its transpose
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    Introduction to orthogonality and change of basis
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    Orthogonal complements
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    Orthogonal complements of the fundamental subspaces
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    Projection onto the subspace
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    Least squares solution
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    Coordinates in a new basis
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    Transformation matrix for a basis
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    Introduction to orthonormal bases and Gram-Schmidt
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    Orthonormal bases
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    Projection onto an orthonormal basis
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    Gram-Schmidt process for change of basis
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    Introduction to Eigenvalues and Eigenvectors
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    Eigenvalues, eigenvectors, eigenspaces
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    Eigen in three dimensions
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    Wrap-up