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What is linear algebra?
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Linear algebra applications
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An enticing start to a linear algebra course!
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How best to learn from this course
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Using MATLAB, Octave, or Python in this course
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Algebraic and geometric interpretations of vectors
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Vector-scalar multiplication
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Vector-vector multiplication: the dot product
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Dot product properties: associative, distributive, commutative
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Code challenge: dot products with matrix columns
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Vector length
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Dot product geometry: sign and orthogonality
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Code challenge: dot product sign and scalar multiplication
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Code challenge: is the dot product commutative?
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Outer product
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Vector cross product
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Vectors with complex numbers
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Hermitian transpose (a.k.a. conjugate transpose)
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Interpreting and creating unit vectors
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Code challenge: dot products with unit vectors
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Dimensions and fields in linear algebra
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Subspaces
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Subspaces vs. subsets
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Span
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Linear independence
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Basis
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Matrix terminology and dimensionality
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A zoo of matrices
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Matrix-scalar multiplication
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Code challenge: is matrix-scalar multiplication a linear operation?
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Transpose
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Complex matrices
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Diagonal and trace
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Code challenge: linearity of trace
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Introduction to standard matrix multiplication
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Four ways to think about matrix multiplication
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Code challenge: matrix multiplication by layering
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Matrix multiplication with a diagonal matrix
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Order-of-operations on matrices
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Matrix-vector multiplication
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2D transformation matrices
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Code challenge: Pure and impure rotation matrices
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Code challenge: Geometric transformations via matrix multiplications
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Code challenge: symmetry of combined symmetric matrices
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Multiplication of two symmetric matrices
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Code challenge: standard and Hadamard multiplication for diagonal matrices
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Code challenge: Fourier transform via matrix multiplication!
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Frobenius dot product
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Matrix norms
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Rank: concepts, terms, and applications
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Computing rank: theory and practice
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Rank of added and multiplied matrices
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Code challenge: reduced-rank matrix via multiplication
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Code challenge: scalar multiplication and rank
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Rank of A^TA and AA^T
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Code challenge: rank of multiplied and summed matrices
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Making a matrix full-rank by "shifting"
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Code challenge: is this vector in the span of this set?
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Course tangent: self-accountability in online learning
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Column space of a matrix
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Column space, visualized in code
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Row space of a matrix
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Null space and left null space of a matrix
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Column/left-null and row/null spaces are orthogonal
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Dimensions of column/row/null spaces
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Example of the four subspaces
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More on Ax=b and Ax=0
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Systems of equations: algebra and geometry
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Converting systems of equations to matrix equations
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Gaussian elimination
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Echelon form and pivots
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Reduced row echelon form
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Code challenge: RREF of matrices with different sizes and ranks
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Matrix spaces after row reduction
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Determinant: concept and applications
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Determinant of a 2x2 matrix
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Code challenge: determinant of small and large singular matrices
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Determinant of a 3x3 matrix
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Code challenge: large matrices with row exchanges
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Find matrix values for a given determinant
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Code challenge: determinant of shifted matrices
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Code challenge: determinant of matrix product
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Matrix inverse: Concept and applications
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Computing the inverse in code
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Inverse of a 2x2 matrix
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The MCA algorithm to compute the inverse
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Code challenge: Implement the MCA algorithm!!
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Computing the inverse via row reduction
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Code challenge: inverse of a diagonal matrix
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Left inverse and right inverse
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One-sided inverses in code
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Proof: the inverse is unique
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Pseudo-inverse, part 1
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Code challenge: pseudoinverse of invertible matrices
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Projections in R^2
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Projections in R^N
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Orthogonal and parallel vector components
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Code challenge: decompose vector to orthogonal components
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Orthogonal matrices
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Gram-Schmidt procedure
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QR decomposition
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Code challenge: Gram-Schmidt algorithm
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Matrix inverse via QR decomposition
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Code challenge: Inverse via QR
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Code challenge: Prove and demonstrate the Sherman-Morrison inverse
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Code challenge: A^TA = R^TR
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Introduction to least-squares
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Least-squares via left inverse
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Least-squares via orthogonal projection
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Least-squares via row-reduction
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Model-predicted values and residuals
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Least-squares application 1
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Least-squares application 2
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Code challenge: Least-squares via QR decomposition
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What are eigenvalues and eigenvectors?
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Finding eigenvalues
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Shortcut for eigenvalues of a 2x2 matrix
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Code challenge: eigenvalues of diagonal and triangular matrices
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Code challenge: eigenvalues of random matrices
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Finding eigenvectors
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Eigendecomposition by hand: two examples
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Diagonalization
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Matrix powers via diagonalization
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Code challenge: eigendecomposition of matrix differences
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Eigenvectors of distinct eigenvalues
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Eigenvectors of repeated eigenvalues
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Eigendecomposition of symmetric matrices
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Eigenlayers of a matrix
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Code challenge: reconstruct a matrix from eigenlayers
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Eigendecomposition of singular matrices
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Code challenge: trace and determinant, eigenvalues sum and product
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Generalized eigendecomposition
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Code challenge: GED in small and large matrices
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Singular value decomposition (SVD)
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Code challenge: SVD vs. eigendecomposition for square symmetric matrices
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Relation between singular values and eigenvalues
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Code challenge: U from eigendecomposition of A^TA
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Code challenge: A^TA, Av, and singular vectors
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SVD and the four subspaces
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Spectral theory of matrices
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SVD for low-rank approximations
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Convert singular values to percent variance
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Code challenge: When is UV^T valid, what is its norm, and is it orthogonal?
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SVD, matrix inverse, and pseudoinverse
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Condition number of a matrix
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Code challenge: Create matrix with desired condition number
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Code challenge: Visualize the normalized quadratic form
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Eigenvectors and the quadratic form surface
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Application of the normalized quadratic form: PCA
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