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**Unformatted text preview: **MATH 304, Fall 2011 Linear Algebra Course outline Part I ( 3 weeks): Elementary linear algebra Systems of linear equations Gaussian elimination process, Gauss-Jordan reduction Matrix algebra Determinants and their properties Part II ( 4.5 weeks): Abstract linear algebra Vector spaces Linear independence of vectors Basis of a vector space and dimension Coordinates, their dependence on change of basis Linear transformations of vector spaces Course outline Part III ( 4 weeks): Advanced linear algebra Orthogonality of vectors, least squares problems Inner product and norm The Gram-Schmidt orthogonalization process Eigenvalues and eigenvectors of matrices Diagonalization of matrices Part IV ( 2 weeks): Some topics in applied linear algebra Matrix exponentials Rotations in space Orthogonal polynomials Fourier series MATH 304 Linear Algebra Part I ( 3 weeks): Elementary linear algebra Systems of linear equations Gaussian elimination, Gauss-Jordan reduction Matrices, matrix algebra Determinants Leons book : Chapters 12 MATH 304 Linear Algebra Lecture 1: Systems of linear equations. Gaussian elimination. Linear equation The equation 2 x + 3 y = 6 is called linear because its solution set is a line in R 2 (and because the powers of x and y are 1: 2 x 1 + 3 y 1 = 6 . The equation x 2 + 3 xy + y 2 = 1 would be called quadratic etc). A solution of the equation is a pair of numbers ( , ) R 2 such that 2 + 3 = 6. For example, (3 , 0) and (0 , 2) are solutions. Alternatively, we can write the first solution as x = 3, y = 0. x y 2 x + 3 y = 6 General equation of a line: ax + by = c , where x , y are variables (called sometime unknowns) and a , b , c are constants (except for the case a = b = 0). Definition. A linear equation in n variables x 1 , x 2 , . . . , x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b , where a 1 , . . . , a n R , and b R are constants....

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