Dual space From Wikipedia, the free encyclopedia Contents 1 Defective matrix 1 1.1 Jordan block ............................................. 1 1.2 Example ............................................... 1 1.3 See also ................................................ 2 1.4 Notes ................................................. 2 1.5 References ............................................... 2 2 Definite quadratic form 3 2.1 Associated symmetric bilinear form .................................. 3 2.2 Example ................................................ 3 2.3 See also ................................................ 3 2.4 References ............................................... 4 3 Delta operator 5 3.1 Examples ............................................... 5 3.2 Basic polynomials ........................................... 6 3.3 See also ................................................ 6 3.4 References .............................................. 6 4 Dependence relation 7 4.1 Examples ............................................... 7 4.2 See also ................................................ 7 5 Determinant 8 5.1 Definition ............................................... 8 5.1.1 2 × 2 matrices ......................................... 9 5.1.2 3 × 3 matrices ......................................... 11 5.1.3 n × n matrices ......................................... 11 5.2 Properties of the determinant ..................................... 13 5.2.1 Multiplicativity and matrix groups .............................. 15 5.2.2 Laplace’s formula and the adjugate matrix .......................... 15 5.2.3 Sylvester’s determinant theorem ............................... 15 5.3 Properties of the determinant in relation to other notions ....................... 16 5.3.1 Relation to eigenvalues and trace ............................... 16 i ii CONTENTS 5.3.2 Cramer’s rule ......................................... 18 5.3.3 Block matrices ........................................ 18 5.3.4 Derivative ........................................... 19 5.4 Abstract algebraic aspects ....................................... 20 5.4.1 Determinant of an endomorphism .............................. 20 5.4.2 Exterior algebra ........................................ 20 5.4.3 Square matrices over commutative rings and abstract properties ............... 21 5.5 Generalizations and related notions .................................. 22 5.5.1 Infinite matrices ........................................ 22 5.5.2 Related notions for non-commutative rings .......................... 22 5.5.3 Further variants ........................................ 22 5.6 Calculation ............................................... 22 5.6.1 Decomposition methods ................................... 22 5.6.2 Further methods ....................................... 23 5.7 History ................................................. 23 5.8 Applications .............................................. 24 5.8.1 Linear independence ..................................... 24 5.8.2 Orientation of a basis ..................................... 24 5.8.3 Volume and Jacobian determinant .............................. 25 5.8.4 Vandermonde determinant (alternant) ............................ 25 5.8.5 Circulants ........................................... 26 5.9 See also ................................................ 26 5.10 Notes ................................................. 26 5.11 References ............................................... 28 5.12 External links ............................................. 28 6 Dieudonné determinant 29 6.1 Properties ............................................... 29 6.2 Tannaka–Artin problem ........................................ 29 6.3 See also ................................................ 29 6.4 References ............................................... 30 7 Dimension (vector space) 31 7.1 Examples ............................................... 31 7.2 Facts ................................................. 31 7.3 Generalizations ............................................ 32 7.3.1 Trace ............................................. 32 7.4 See also ................................................ 32 7.5 Notes ................................................. 33 7.6 References .............................................. 33 7.7 External links ............................................. 33 CONTENTS iii 8 Direct sum of modules 34 8.1 Construction for vector spaces and abelian groups .......................... 34 8.1.1 Construction for two vector spaces .............................. 34 8.1.2 Construction for two abelian groups ............................. 35 8.2 Construction for an arbitrary family of modules ........................... 35 8.3 Properties ............................................... 36 8.4 Internal direct sum .......................................... 37 8.5 Universal property .......................................... 37 8.6 Grothendieck group .......................................... 37 8.7 Direct sum of modules with additional structure ........................... 37 8.7.1 Direct sum of algebras .................................... 37 8.7.2 Direct sum of Banach spaces ................................. 38 8.7.3 Direct sum of modules with bilinear forms .......................... 38 8.7.4 Direct sum of Hilbert spaces ................................. 38 8.8 See also ................................................ 39 8.9 References .............................................. 39 9 Direction vector 41 9.1 Parametric equation for a line ..................................... 41 9.2 Generative versus predicate forms ................................... 41 9.3 Predicate form of 2D line equation .................................. 41 9.4 See also ................................................ 42 9.5 External links ............................................. 42 10 Dot product 43 10.1 Definition ............................................... 43 10.1.1 Algebraic definition ...................................... 43 10.1.2 Geometric definition ..................................... 44 10.1.3 Scalar projection and first properties ............................. 44 10.1.4 Equivalence of the definitions ................................. 45 10.2 Properties ............................................... 46 10.2.1 Application to the cosine law ................................. 47 10.3 Triple product expansion ....................................... 47 10.4 Physics ................................................. 48 10.5 Generalizations ............................................ 49 10.5.1 Complex vectors ....................................... 49 10.5.2 Inner product ......................................... 49 10.5.3 Functions ........................................... 49 10.5.4 Weight function ........................................ 50 10.5.5 Dyadics and matrices ..................................... 50 10.5.6 Tensors ............................................ 50 10.6 See also ................................................ 50 iv CONTENTS 10.7 References ............................................... 50 10.8 External links ............................................. 50 11 Dual basis 51 11.1 A categorical and algebraic construction of the dual space ...................... 51 11.2 Existence and uniqueness ....................................... 51 11.3 Introduction .............................................. 52 11.4 Examples ............................................... 52 11.5 See also ................................................ 53 11.6 References ............................................... 53 12 Dual basis in a field extension 54 13 Dual norm 55 13.1 Examples ............................................... 55 13.2 Notes ................................................. 55 13.3 References .............................................. 56 14 Dual number 57 14.1 Linear representation ......................................... 57 14.2 Geometry ............................................... 57 14.2.1 Cycles ............................................. 58 14.3 Algebraic properties .......................................... 58 14.4 Generalization ............................................. 58 14.5 Differentiation ............................................. 58 14.6 Superspace ............................................... 59 14.7 Division ................................................ 59 14.8 Projective line ............................................. 60 14.9 See also ................................................ 60 14.10Notes and references ......................................... 60 14.11Further reading ............................................ 61 15 Dual space 62 15.1 Algebraic dual space ......................................... 62 15.1.1 Finite-dimensional case ................................... 62 15.1.2 Infinite-dimensional case ................................... 63 15.1.3 Bilinear products and dual spaces .............................. 64 15.1.4 Injection into the double-dual ................................ 64 15.1.5 Transpose of a linear map .................................. 65 15.1.6 Quotient spaces and annihilators ............................... 65 15.2 Continuous dual space ........................................ 66 15.2.1 Examples ........................................... 67 15.2.2 Transpose of a continuous linear map ............................ 67 CONTENTS v 15.2.3 Annihilators ........................................