Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Unit 3: Linear Systems, Vector Spaces Definition. [–] A set V is called a linear space or a vector space over R if there are two operations + and · with the following properties • v 1 , v2 ∈ V ⇒ v 1 + v 2 ∈ V • v 1 + v 2 = v2 + v 1 • v1 + (v2 + v3) = (v1 + v2) + v3 • There is an element 0V ∈ V with 0V + v = v + 0V = v • There is an element −v ∈ V with v + (−v) = 0V C. Führer: FMN050/FMNF01-2011 47 3.1: Vector Spaces (Cont.) • α∈R⇒α·v ∈V • (α + β) · v = α · v + β · v • α · (β · u) = (αβ) · u • α · (v1 + v2) = α · v1 + α · v2 • 1·v =v One can then easily show 0 · v = 0 and −1 · v = −v. The elements v of V are called vectors, the elements of R scalars. C. Führer: FMN050/FMNF01-2011 48 3.2: Example of Vector Spaces In linear Algebra: R, the nullspace and the column space of a matrix In this course: • space of all n × m matrices • space of all polynomials of degree n: P n • space of all continuous functions C[a, b] • space of all functions with a continuous first derivative C 1[a, b] • .... C. Führer: FMN050/FMNF01-2011 49 3.3: Example of Vector Spaces Not a vector space: The set of all polynomials of degree n which have the property p(2) = 5. C. Führer: FMN050/FMNF01-2011 50 3.4: Basis, Coordinates One describes a certain element in a vector space by giving its coordinates in a given basis. (Recall these corresponding definitions from linear algebra). The number of basis vectors determines the dimension of the vector space. C[a, b] is a vector space with an infinite dimension, P n has finite dimension. C. Führer: FMN050/FMNF01-2011 51 3.5: Norms First we recall properties of the absolute value (absolut belopp) of a real number: � v if v ≥ 0 v ∈ R �→ |v| = −v if v < 0 • |v| ≥ 0 and |v| = 0 ⇔ v = 0 • |λv| = |λ||v| • |u + v| ≤ |u| + |v| C. Führer: FMN050/FMNF01-2011 52 3.6: Vector Norms We generalize the definition of the absolute value of a real number to norms of vectors and matrices (later also of functions): Definition. [2.6] V a linear space , a function � · � : V → R is called a norm if for all u, v ∈ V and all λ ∈ R: • �v� ≥ 0 and �v� = 0 ⇔ v = 0 (Positivity) • �λv� = |λ|�v� (Homogenity) • �u + v� ≤ �u� + �v� (Triangular inequality) C. Führer: FMN050/FMNF01-2011 53 3.7: Examples Examples for norms in Rn: • 1-norm �v�1 = n � i=1 |vi| • 2-norm (Euclidean norm) �v�2 = • ∞-norm n �� i=1 � 2 1/2 vi max |vi| i=1:n C. Führer: FMN050/FMNF01-2011 54 3.8: Unit Circle The unit circle is the set of all vectors of norm 1. inf 1 1 0 2 -1 -1 0 1 C. Führer: FMN050/FMNF01-2011 55 3.9: Convergence Theorem. [-] If dim V < ∞ and if � · �p and � · �q are norms in V, then there exist constants c, C such that for all v ∈ V c�v�q ≤ �v�p ≤ C�v�q Norms in finite dimensional spaces are equivalent. Sequences convergent in one norm are convergent in all others. C. Führer: FMN050/FMNF01-2011 56 3.10: Matrix norms A matrix defines a linear map. Definition. [2.10] Let � · � be a given vector norm. The corresponding (subordinate) matrix norm is defined as �Av� �A� = max v∈Rn \{0} �v� I.e. the largest relative change of a vector, when mapped by A. C. Führer: FMN050/FMNF01-2011 57 3.11: Matrix norms Example: A = � � 2 1 1 1 2 A 0 -2 -1 0 1 -1 0 1 �A� = 2.6180 C. Führer: FMN050/FMNF01-2011 58 3.12: How to compute matrix norms Matrix norms are computed by applying the following formulas: 1-norm (Th. 2.8): �A�1 = maxj=1:n �n i=1 |aij | ∞-norm (Th. 2.7): �A�1 = maxi=1:n 2-norm (Th. 2.9): �A�2 = maxi=1:n �n � j=1 |aij | maximal column sum maximal row sum λi(ATA) where λi(ATA) is the ith eigenvalue of ATA. C. Führer: FMN050/FMNF01-2011 59