linear-algebra - NumericalLinearAlgebra ChrisRambicure...

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    Numerical Linear Algebra Chris Rambicure Guojin Chen Christopher Cprek
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    WHY USE LINEAR  ALGEBRA? 1) Because it is applicable in many  problems…. 2)…And it’s usually easier than calculus
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    TRUE “Linear algebra has become as basic and as applicable as calculus,and fortunately it is easier.” -Gilbert Strang Calculus
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  HERE COME THE BASICS…
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    SCALARS What you’re used to dealing with Have magnitude, but no direction
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    VECTORS Represent both a magnitude and a  direction Can add or subtract, multiply by scalars,  or do dot or cross products
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    THE MATRIX It’s an mxn array Holds a set of numerical values Especially useful in solving certain types  of equations Operations: Transpose, Scalar Multiply,  Matrix Add, Matrix Multiply
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    EIGENVALUES You can choose a matrix A, a vector x,  and a scalar x so that Ax = sx, meaning  the matrix just scales the vector X in this case is called an eigenvector,  and s is its eigenvalue
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    CHARACTERISTIC  EQUATION det(M-tI) = 0 M: the matrix I: the identity t: eigenvalues
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    CAYLEY-HAMILTON  THEOREM IF  AND THEN p(A) = 0, meaning A satisfies its  characteristic equation
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    A Couple Names, A Couple  Algorithms
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    IN THE BEGINNING… (Grassmann’s Linear Algebra) Grassmann is considered to be the  “father” of linear algebra Developed the idea of a linear algebra  in which the symbols representing  geometric objects can be manipulated Several of his operations: the interior  product, the exterior product, and the  multiproduct
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    What’s a Multiproduct  Equation Look Like? δ 1 ⊗δ 2  +  δ 1 2  = 0 The multiproduct has many uses,  including scientific, mathematic, and  industrial Got updated by William Clifford
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    TO GRASSMAN’S  EQUATION δ 1 ⊗δ 2  +  δ 1 2  = 2k ij The 2k ij  is what’s referred to as  Kronecker’s Symbol Both of these equations are used for  Quantum Theory Math
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    VECTOR SPACE Another idea which is kind of tied with  Grassman Vector Space refers to some set of  vectors that contains the origin It is usually infinite Subspace is a subset of vector space.   It, of course, is also vector space
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    Cholesky Decomposition Algorithm developed by Arthur Cayley Takes a matrix and factors it into a  triangular matrix times its transpose A=R’R Useful for matrix applications Becomes even more worthwhile in  parallel
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    HOW TO USE LINEAR  ALGEBRA FOR PDE’S You can use matrices and vectors to  solve partial differential equations For equations with lots of variables,  you’ll wind up with really sparse  matrices Hence, the project we’ve been working  on all year
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    BIBLIOGRAPHY “Hermann Grassmann.” Online.  http://members.fortunecity.com/johnhays/grassmann.h
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linear-algebra - NumericalLinearAlgebra ChrisRambicure...

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