# Overhead-Part 02-Linear Alg_Rev18 - REVISED 10/13/2009 II....

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II - Review of Linear Algebra © Oscar D. Crisalle 2005 - 2009 1 II. Review of Linear Algebra ECH 6326 Prof. Oscar D. Crisalle

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II - Review of Linear Algebra © Oscar D. Crisalle 200 5- 2009 Contents Real and complex numbers Linear combination of vectors Matrices Partitions, transpositions, determinants Matrix rank Matrix inversion The adjoint matrix (Cramer’s rule) Rank and null space Rank and homogeneous equations Left an right pseudoinverses Left and right inverses Solution to homogeneous equations Solution to nonhomogeneous equations Eigenvalues and eigenvectors Classification of eigenvalues
II - Review of Linear Algebra © Oscar D. Crisalle 2005 - 2009 1 Real and Complex Numbers Scalars R Set of real numbers C = { σ + i ω : σ , ω R } Set of complex numbers Z = { . .., -2, -1, 0, 1, 2, . .. } Set of integers N = { 0, 1, 2, . .. } Set of natural numbers

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II - Review of Linear Algebra © Oscar D. Crisalle 2005 - 2009 2 Vectors Set of ordered n-tuples (Cartesian product space) ! n = { x 1 , x 2 , x 3 , " , x n } : x i ! ! , i = 1,2,. .., n { } = ! " ! " ! " " " ! Column vector x ! ! n " x = x 1 x 2 " x n # \$ % % % % % & ( ( ( ( ( ! ! n x isa column vectorin ! n Row vector x ! ! n " x = x 1 x 2 " x n # \$ % & ! ! n x row ! n Better notation for row vectors: x ! ! 1 " n (one row, n columns)
II - Review of Linear Algebra © Oscar D. Crisalle 2005 - 2009 3 Linear Combination of Vectors Let ! 1 , ! 2 , ! 3 , ! , ! n be a set of (real or complex) scalars Definition - Linear combination of vectors ! 1 x 1 + ! 2 x 2 + ! 3 x 3 + ! + ! n x n Definition - Linear independence of vectors . A set of vectors { x 1 , x 2 , x 3 , ! , x n } is said to be linearly independent if the only solution to the equation ! 1 x 1 + ! 2 x 2 + ! 3 x 3 + ! + ! n x n = 0 is the trivial solution ! 1 = ! 2 = ! 3 = ! = ! n = 0 . Otherwise, the set of vectors is said to be linearly dependent . Two collinear vectors are linearly dependent

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II - Review of Linear Algebra © Oscar D. Crisalle 2005 - 2009 4 α l = 2, α 2 = -1 2 x 1 – 1 x 2 = 0 Two noncollinear vectors are linearly independent only α l = α 2 = 0 can make α 1 x 1 + α 2 x 2 = 0 Three noncollinear vectors lying on the same plane are linearly dependent α l = 1, α 2 = 1, α 3 = 1 1 x 1 + 1 x 2 + 1 x 3 = 0 Definition . An n -tuplet of scalars { ! 1 , ! 2 , ! , ! j , ! , ! n } is said to be a trivial set if ! 1 = ! 2 = ! = ! j = ! = ! n = 0 . If at least one element of the set is nonzero, then the n -tuplet is said to be nontrivial .