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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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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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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