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Week I
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August 25–29
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Linear equations; systems of linear equations; matrices; row operations; echelon and reduced echelon forms; vectors; properties of vectors
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Week II
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September 2–5
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Vector addition and scalar multiplication; the zero vector; the properties of vectors in Rn; linear combinations; the span; the matrix equation, Ax=b; index notation
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Week III
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September 8–12
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The identity matrix; the Kronecker delta; linearity of the action matrices on vectors; homogeneous sets of linear equations; parametric solutions; inhomogeneous equations; linear independence; small sets of vectors; linear transformations
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Week IV
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September 15–19
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Linear transformations; domains, codomains, images, and ranges; matrix transformations; linearity; the matrix of a linear transformation; dilations; rotations; injective, surjective, and bijective transformations
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Week V
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September 22–26
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Matrix algebra; where the indices go and what they mean; diagonal matrices; the zero matrix; adding matrices; scalar multiplication; matrix multiplication; functions of matrices; the transpose
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Week VI
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Sept 29–Oct 3
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The dot product and the transpose; properties of the transpose; orthogonal matrices; Hermitian conjugation; Hermitian matrices; inverting matrices; AA–1=A–1A=I; singular and nonsingular matrices; the determinant of a matrix; properties of inveritble matrices; row operations as matrices; an algorithm for find the inverse of a matrix
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Week VII
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October 6–10
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The invertible matrix theorem; invertible linear transformations; partitioned matrices and blocks; examples of partitions; blocks and matrix operations; the ‘column-row’ expansion for the product of two matrices; block-diagonal matrices; inverting a block upper triangular matrix
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Week VIII
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October 13–16
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The factorization of matrices; an algorithm for finding the LU factorization of a matrix; subspaces, their definition and their properties; matrices and subspaces; the column space of a matrix; the null space of a matrix; the kernel
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Week IX
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October 20–24
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Bases and their definition; the basis of the column space of a matrix; coordinates; uniqueness of the coordinates in a basis; isomorphisms; dimension of a subspace; the rank of a matrix; the rank theorem; invertible matrix theorem (continued); changing bases
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Week X
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October 27–31
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Changing matrices; determinants; how to find a 3×3 determinant; cofactors; how to calculate the determinant of an n×n matrix; the determinant of a triangular matrix; determinants of row operations; column operations; det(AB)= det(A)det(B)
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Week XI
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November 3–7
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Linearity of the determinant; det(AB)= det(A)det(B); Cramer’s rule; inverting matrices using Cramer’s rule; the adjugate of a matrix; how areas and volumes change under a linear transformation; Markov processes; probability vectors; stochastic matrices; state vectors; Markov chains; evolving into the distant future
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Week XII
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November 10–14
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Regular stochastic matrices; convergence of a Markov chain to the steady-state vector; eigenvalues and eigenvectors, Ax = λx; how to find an eigenvector as the solution to a homogeneous equation, [A–λI]x = 0; eigenspaces; the eigenvalues of triangular matrices; zero as an eigenvalue; eigenvalues and linear independnce; eigenvectors and difference equations; the characteristic equation, det(A–λI) = 0; the characteristic polynomial
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Week XIII
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November 17–21
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Roots of polynomials; multiplicity of eigenvalues; similar matrices, B=P–1AP; similarity transformations; diagonalising matrices; A=PDP–1; the diagonalisation theorem; an algorithm for how to diagonalise a matrix; the eigenvectors are the columns of the change of basis matrix
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Week XIV
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November 24–25
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Diagonalising symmetric matrices; orthogonal vectors; diagonalising with an orthogonal matrix
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Week XV
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December 1–5
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Projection matrices; orthogonal subspaces; projections onto subspaces; the spectral decomposition of a matrix; the Gram-Schmidt procedure; least-squares
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