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M. Sami Fadali EBME Dept., UNR Metric space. Linear vector space. Linear independence & bases. Normed spaces. 1 2 Union Intersection Cartesian Product Pair (nonempty set, metric) = Metric (distance measure) axioms (i) (ii) (iii) Show that : 3 4 Vector space over with two functions is a nonempty set Multiplicative identity Distributive multiplication Associative multiplication Commutative addition Associative addition Additive zero Additive inverse (real or complex vector space) 5 6 Vector space is closed under addition Vector space is closed under scalar multiplication Can define an inner product for 7 8 =Class of continuous function over the closed interval exists Vector space is closed under addition linearly independent set linearly dependent set Vector space is closed under scalar multiplication not all zero. 9 Set of linearly independent vectors 10 Any vector , can be written as the linear combination Any nonempty subset of closed under addition and scalar multiplication. Example: The set of vectors Dimension of vector space: number of basis vectors. = subspace of 11 12 (i) iff (ii) (iii) + Show norm is nonnegative A vector space with a norm Can be regarded as metric spaces with the metric Satisfies zero iff , symmetry, and the triangle inequality for metrics + 13 Make norm 14 a normed vector space with / Holder’s Inequality Let One-norm Two-norm (Euclidean) Infinity-norm Equivalent , then Minkowski Inequality , then Let / 15 16 Matrix Identities Inverse Transpose Inverse Transpose Algebraic multiplicity = number of eigenvalue repetitions of the Geometric multiplicity = number of linearly independent eigenvectors for the eigenvalue. Diagonalizable matrix: , always true for distinct eigenvalues. 17 18 Positive Definite Positive Semidefinite Negative Definite Negative Semidefinite Assume symmetric matrix w.l.o.g. 19 Indefinite Matrix for nonzero . has no definite sign 20 For a symmetric matrix Proof = The results follow by considering the different cases for the signs of the eigenvalues. e.g. positive definite for 21 Neighborhood of a point s.t. : Set . Limit Point of : every neighvorhood of contains a point .Note need not be in Interior point neighborhood 22 : There exists a . 23 Open Set: Every point in the set is an interior point. Closed Set: Contains all its limit points. Complement of set If is closed then versa. Bounded Set : is open and vice 24 Neighborhood: open ball in Closed Set Bounded Set : Compact Set: closed and bounded. Convex Set : Convergent Sequence of vectors to a limit : → Limit of → Cauchy Sequence in a metric space : for every real number there is an integer s.t. Not Convex Convex 25 26 Complete Metric Space : every Cauchy sequence converges (limit is in the space), i.e. for every Cauchy s.t. as sequence . Banach Space: complete normed space. Hilbert Space: complete inner product space. 27 Let and be abstract sets. Function is a set of ordered pairs in the Cartesian product and if , then Only one element in (range) assigned to (domain) Injective (one-one) Surjective (onto) s.t. Bijective: injective and surjective. 28 and Function metric spaces s.t. is continuous on : points in For normed spaces: Uniform Continuity: Linear Operator (transformation) s.t. is continuous at all Bounded Operator and Operator Norm independent of 29 All linear operators Induced Matrix Norms are of the form real matrix 30 Function interval defined in an open and the derivative _|Å exists. Function set , → defined in an open and the derivative _|Å exists. 31 → 32 A function is continuously differentiable at a point if at It is continuous. Its partial derivatives are continuous. A function is said to be of class if the , exist and are derivatives continuous. = continuously differentiable. Lipschitz Condition Lipschitz on an open set Globally Lipschitz if Contraction Mapping 33 Sufficient Condition for Lipschitz : is If a function continuously differentiable on an open set , then it is locally Lipschitz on . Necessary Condition for Lipschitz : uniformly continuous function 34 Solution has a known form for linear case. No general form for the solution in the nonlinear case. Questions: ◦ Does a solutions exist? ◦ Is the solution unique? 35 36 Nonlinear differential eqn. Nonlinear differential eqn. (i) piecewise-continuous in (ii) Satisfies the Lipschitz Condition (i) piecewise-continuous in (ii) Satisfies the Lipschitz Condition The eqn. has a unique solution over s.t. the eqn. has a unique solution in the interval 37 38