- Prerequisite:high school mathematics
- Recommended for: freshmen
- Introduction:
This course provides a first introduction into the origin, theory and application of Linear Algebra.
- Matrices and Linear System<ol>Matrices and their algebra, solving systems of linear equations, inverse of square matrices, homogeneous linear systems, computation of inverse matrix, Gaussian elimination, existence and uniqueness of solutions, reduced row ech-elon form, row rank and column rank, nullity</ol>
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Vector Spaces<ol>Euclidean spaces, general vector spaces, linear subspaces, independence, span, basis and dimension, coordinatization of vectors</ol>
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Linear Transformation<ol>Invariant subspace, kernel, range, rank-nullity theorem, matrix representation and change of basis, similarity</ol>
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Determinant<ol>Areas, volumes, cross product, properties of determinant, computation of determinant, Cramer's rule</ol>
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Eigenvalues and Eigenvectors<ol>Characteristic polynomials, algebraic and geometric multiplicities, eigenspaces, diagonalizations and triangularization, invariant subspaces</ol>
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Inner Product Space<ol>Inner product, norm, adjoint of a matrix, bilinear forms*</ol>
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Orthogonality and Projections<ol>Orthogonal matrices, Gram-Schmidt process, mutually perpendicular vectors and subspaces, orthogonal projection.</ol>
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Quadratic Forms<ol>Diagonalization of quadratic forms, positive and negative definite quadratic forms, applications to extrema</ol>
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Special Matrices<ol>Tridiagonal, symmetric, projection, unitary, Hermitian, normal, nilpotent matrices and their properties</ol>
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Complex Scalars and Vector Spaces<ol>Matrices and vector spaces over complex scalars, complex Euclidean inner product, complex subspaces*, Jordan canonical forms*, minimal polynomial*</ol>
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Applications*<ol>Markov chain, Fibonacci sequence, solving system of ODE, method of least squares, etc.</ol>
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Additional topics as time permits<ol>Direct sum, direct sum decomposition of vector spaces, direct sum decomposition of linear operators, dual spaces</ol>
- Linear Algebra, by J. B. Fraleigh and R. A. Beaurgard
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Linear Algebra, by S. H. Friedberg, A. J. Insel, and L. E. Spence
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Linear Algebra, by K. Hoffman and R. Kunze
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Linear Algebra and its Applications, by G. Strang
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Differential Equations, Dynamical Systems, and Linear Algebra, by M. Hirsch and S. Smale
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