StudyGuide_short - E3101: A study guide and review, Short...

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E3101: A study guide and review, Short Form Version 1.3 Marc Spiegelman December 5, 2007 1 The Short-form Okay, here is the entire study guide in compact form Solving A x = b for A R n × n Equation: A x = b Algorithm: Gaussian Elimination [ A b ] [ U c ] and backsubstitute Factorization: PA = LU (or PA = LDU or PA = LDL T if A T = A . Also A = CC T (Cholesky factorization for SPD matrices). The Matrix Inverse A - 1 Definition: AA - 1 = A - 1 A = I , solves x = A - 1 b . Existence: A - 1 exists iff Gaussian Elimination produces n pivots (i.e. n linearly inde- pendent columns). Uniqueness: if A is invertible, A - 1 is unique and x = A - 1 b is unique Algorithm: Gauss-Jordan Elimination [ A I ] [ U E d ] [ D E u E d ] [ I D - 1 E u E d ] = [ I A - 1 ] Product Rules For Matrices General AB : inner products must agree. In general AB 6 = BA . Inverse: ( AB ) - 1 = B - 1 A - 1 (if A , B both square and invertible) Transpose: ( AB ) T = B T A T (all A and B such that AB exists. Determinant: | AB | = | A || B | 1
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E3101: Study Guide 2007 2 For vectors – x T y : inner product, dot product, maps R n R – x T y = y T x – x T x = || x || 2 , where || x || is length of x – xy T : outer product. Is a rank 1 matrix. if x R m and y R n then xy T R m × n – xy T 6 = yx T General Solutions of A x = b for A R m × n Algorithm: Gauss-Jordan Elimination [ A b ] [ R d ] where R is reduced row echelon form Identify rank of A (number of pivot columns) and label pivot and free columns. Check existence of solution ( d C ( R ) implies b C ( A ) ). Solve R x p = d for the particular solution x p (i.e. combination of pivot columns and no free columns that add to d ). Find special solutions as basis of N ( R ) = N ( A ) . General solution is x = x p + N c if b C ( A ) Note: x p not usually entirely in C ( A T ) (i.e. x p 6 = x + ). x + = A + b = A + A x p The four fundamental subspaces of A R m × n Definition of Basis: a minimum set of linearly independent vectors that span a vector space or subspace. Definition of Dimension: the number of basis vectors for any subspace. The four subspaces of a matrix A which is m × n with rank r Name Symbol Dimension Basis Row Space C ( A T ) R n r linearly independent rows of R Null Space N ( A ) R n n - r special solutions of A x = 0 Column Space C ( A ) R m r linearly independent columns of A Left Null Space N ( A T ) R m m - r special solutions of A T x = 0
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This note was uploaded on 04/08/2008 for the course APPLIED MA 3101 taught by Professor Spiegelman during the Spring '08 term at Columbia.

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StudyGuide_short - E3101: A study guide and review, Short...

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