Linear_Algebra_Notes_Packard

Linear_Algebra_Notes_Packard - Linear Algebra 1 Set...

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Linear Algebra Preliminaries 1. Set notation 2. Fields, vector spaces, normed vector spaces, inner product spaces 3. More notation 4. Vectors in R n , C n , norms 5. Matrix Facts (determinants, inversion formulae) 6. Normed vector spaces, inner product spaces 7. Linear transformations 8. Matrices, matrix multiplication as linear transformation 9. Induced norms of matrices 10. Schur decomposition of matrices 11. Symmetric, Hermitian and Normal matrices 12. Positive and Negative definite matrices 13. Singular Value decomposition 14. Hermitian square roots of positive semidefinite matrices 15. Schur complements 16. Matrix Dilation, Parrott’s theorem 17. Completion of Squares ME 234, UC Berkeley, Spring 2003, Packard 65
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Some Notation Sets, functions 1. R is the set of real numbers. C is the set of complex numbers. 2. N is the set of integers. 3. The set of all n × 1 column vectors with real number entries is denoted R n . The i ’th entry of a column vector x is denoted x i . 4. The set of all n × m rectangular matrices with complex number entries is denoted C n × m . The element in the i ’th row, j ’th column of a matrix M is denoted by M ij , or m ij . 5. Set notation: (a) a A is read: “ a is an element of A (b) X Y is read: “ X is a subset of Y (c) If A and B are sets, then A × B is a new set, consisting of all ordered-pairs drawn from A and B , A × B := { ( a, b ) : a A, b B } (d) The expression {A : B} is read as: “The set of all insert expression A such that insert expression B .” Hence x R 3 : 3 X i =1 x 2 i 1 is the ball of radius 1, centered at the origin, in 3-dimensional euclidean space. 6. The notation f : X Y implies that X and Y are sets, and f is a function mapping X into Y ME 234, UC Berkeley, Spring 2003, Packard 66
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Fields A field consists of: a set F (which must contain at least 2 elements) and two operations, addition (+) and multiplication ( · ), each map- ping F × F → F . Several axioms must be satisfied: For every a, b ∈ F , there corresponds an element a + b ∈ F , the addition of a and b . For all a, b, c ∈ F , it must be that a + b = b + a ( a + b ) + c = a + ( b + c ) There is a unique element θ ∈ F (or 0 F , θ F , or just 0) such that for every a ∈ F , a + θ = a . Moreover, for every a ∈ F , there is a unique element labled - a such that a + ( - a ) = θ . For every a, b ∈ F , there corresponds an element a · b ∈ F , the multiplication of a and b . For every a, b, c ∈ F a · b = b · a a · ( b · c ) = ( a · b ) · c. There is a unique element 1 F ∈ F (or just 1) such that for every a ∈ F , 1 · a = a · 1 = a . Moreover, for every a ∈ F , a 6 = θ , there is a unique element, labled a - 1 ∈ F such that a · a - 1 = 1 F . For every a, b, c ∈ F , a · ( b + c ) = a · b + a · c Example: The real numbers R , the complex numbers C , and the rational numbers Q are three examples of fields.
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