linear mappings
... here exists T-1 A(V) such that TT-1=T-1T = 1 Now T is invertible if and only if it is one-one and onto. Thus in particular, if T is invertible then ony 0V can map into itself, i.e. T is nonsingular. On the other hand, suppose T is nonsingular, i.e. Ker T = {0}. Recall that T is also one-one. Moreo ...
... here exists T-1 A(V) such that TT-1=T-1T = 1 Now T is invertible if and only if it is one-one and onto. Thus in particular, if T is invertible then ony 0V can map into itself, i.e. T is nonsingular. On the other hand, suppose T is nonsingular, i.e. Ker T = {0}. Recall that T is also one-one. Moreo ...
Chapter 4 Vector Spaces
... Proof. We have to prove this theorem using the definition 4.2.1. Other than that, the proof will be similar to theorem 4.1.5. To prove (1), write w = 0v. We have w = 0v = (0 + 0)v = 0v + 0v = w + w (by distributivityP rop.(2c)). Add −w to both sides w + (−w) = (w + w) + (−w) By (1e) of 4.2.1, we hav ...
... Proof. We have to prove this theorem using the definition 4.2.1. Other than that, the proof will be similar to theorem 4.1.5. To prove (1), write w = 0v. We have w = 0v = (0 + 0)v = 0v + 0v = w + w (by distributivityP rop.(2c)). Add −w to both sides w + (−w) = (w + w) + (−w) By (1e) of 4.2.1, we hav ...
Universal Drinfeld-Sokolov Reduction and Matrices of Complex Size
... This paper is devoted to a relation between these two approaches to the classical Wn –algebras (called also Adler–Gelfand–Dickey or higher KdV -structures), natural infinitedimensional quadratic Poisson structures on differential operators of nth order. The noncommutative Hamiltonian reduction (see[ ...
... This paper is devoted to a relation between these two approaches to the classical Wn –algebras (called also Adler–Gelfand–Dickey or higher KdV -structures), natural infinitedimensional quadratic Poisson structures on differential operators of nth order. The noncommutative Hamiltonian reduction (see[ ...
Article - Archive ouverte UNIGE
... 3. Quadratic Lie algebras In this Section, we specialize Theorem 2.1 to Lie algebras with an invariant quadratic form. We show that the exponentials in the Clifford and exterior algebras satisfy natural differential equations. In the following Section the dynamical Yang-Baxter equation emerges as a ...
... 3. Quadratic Lie algebras In this Section, we specialize Theorem 2.1 to Lie algebras with an invariant quadratic form. We show that the exponentials in the Clifford and exterior algebras satisfy natural differential equations. In the following Section the dynamical Yang-Baxter equation emerges as a ...
3-Calabi-Yau Algebras from Steiner Systems
... As shown by Ginzburg [Gin06], numerous concrete examples of Calabi-Yau algebras are found “in nature”, and in most cases they arise as a certain quotient of the free associative algebra in the way described below. We refer the reader to Bocklandt’s work in [Boc08] for a deeper approach on the subjec ...
... As shown by Ginzburg [Gin06], numerous concrete examples of Calabi-Yau algebras are found “in nature”, and in most cases they arise as a certain quotient of the free associative algebra in the way described below. We refer the reader to Bocklandt’s work in [Boc08] for a deeper approach on the subjec ...
Vector coordinates, matrix elements and changes of basis
... where P is the matrix whose columns are the eigenvectors of A and D is the diagonal matrix whose diagonal elements are the eigenvalues of A. Thus, we have succeeded in diagonalizing an arbitrary semi-simple matrix. If the eigenvectors of A do not span the vector space V (i.e., A is defective), then ...
... where P is the matrix whose columns are the eigenvectors of A and D is the diagonal matrix whose diagonal elements are the eigenvalues of A. Thus, we have succeeded in diagonalizing an arbitrary semi-simple matrix. If the eigenvectors of A do not span the vector space V (i.e., A is defective), then ...
Linear Algebra Done Right, Second Edition
... multiplication on Fn is associative, and scalar multiplication by 1 acts as a multiplicative identity should. Finally, addition and scalar multiplication on Fn are connected by distributive properties. We will define a vector space to be a set V along with an addition and a scalar multiplication on V ...
... multiplication on Fn is associative, and scalar multiplication by 1 acts as a multiplicative identity should. Finally, addition and scalar multiplication on Fn are connected by distributive properties. We will define a vector space to be a set V along with an addition and a scalar multiplication on V ...
Chapter 1
... multiplication on Fn is associative, and scalar multiplication by 1 acts as a multiplicative identity should. Finally, addition and scalar multiplication on Fn are connected by distributive properties. We will define a vector space to be a set V along with an addition and a scalar multiplication on V ...
... multiplication on Fn is associative, and scalar multiplication by 1 acts as a multiplicative identity should. Finally, addition and scalar multiplication on Fn are connected by distributive properties. We will define a vector space to be a set V along with an addition and a scalar multiplication on V ...
Tense Operators on Basic Algebras - Phoenix
... Abstract The concept of tense operators on a basic algebra is introduced. Since basic algebras can serve as an axiomatization of a many-valued quantum logic (see e.g. Chajda et al. in Algebra Univer. 60(1):63–90, 2009), these tense operators are considered to quantify time dimension, i.e. one expres ...
... Abstract The concept of tense operators on a basic algebra is introduced. Since basic algebras can serve as an axiomatization of a many-valued quantum logic (see e.g. Chajda et al. in Algebra Univer. 60(1):63–90, 2009), these tense operators are considered to quantify time dimension, i.e. one expres ...
PDF
... continuous functions with compact support on a groupoid G is made into a *-algebra whose multiplication is the convolution, and that is also endowed with the smallest C ∗ –norm which makes its representations continuous, as shown in ref.[?]. Furthermore, for this convolution to be defined, one needs ...
... continuous functions with compact support on a groupoid G is made into a *-algebra whose multiplication is the convolution, and that is also endowed with the smallest C ∗ –norm which makes its representations continuous, as shown in ref.[?]. Furthermore, for this convolution to be defined, one needs ...
Quotient spaces defined by linear relations
... strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz ...
... strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz ...
Some results on the existence of division algebras over R
... that the zero algebra, the Real numbers and the Complex numbers form division algebras of respective dimension 0, 1 and 2 over R. In the rest of the chapter, it is proven that furthermore, the Hamilton numbers (otherwise known as the Quaternions) form a 4-dimensional division algebra over R, and tha ...
... that the zero algebra, the Real numbers and the Complex numbers form division algebras of respective dimension 0, 1 and 2 over R. In the rest of the chapter, it is proven that furthermore, the Hamilton numbers (otherwise known as the Quaternions) form a 4-dimensional division algebra over R, and tha ...
Appendix B Lie groups and Lie algebras
... of continuous maps M → M , depending sometimes on extra (e.g. smooth) structures; these details will not be integral to our discussion, so we will suppress them. It will at least be clear in all the examples of interest that the groups under consideration can be regarded as topological groups. The s ...
... of continuous maps M → M , depending sometimes on extra (e.g. smooth) structures; these details will not be integral to our discussion, so we will suppress them. It will at least be clear in all the examples of interest that the groups under consideration can be regarded as topological groups. The s ...
Determination of the Differentiably Simple Rings with a
... simplewitha minimalideal,and that if C containsa subfieldE such that A/N is centralover E and if A (as an E algebra) containsa subalgebraS -A/N with S + N = A then A S 0 EC; moreoverit is essentiallyshown that such an S exists if A is d-simpleforsomederivationd. A proofof the commutativeassociative ...
... simplewitha minimalideal,and that if C containsa subfieldE such that A/N is centralover E and if A (as an E algebra) containsa subalgebraS -A/N with S + N = A then A S 0 EC; moreoverit is essentiallyshown that such an S exists if A is d-simpleforsomederivationd. A proofof the commutativeassociative ...
INFINITESIMAL BIALGEBRAS, PRE
... An infinitesimal bialgebra (abbreviated ǫ-bialgebra) is a triple (A, µ, ∆) where (A, µ) is an associative algebra, (A, ∆) is a coassociative coalgebra, and ∆ is a derivation (see Section 2). We write ∆(a) = a1⊗a2 , omitting the sum symbol. Infinitesimal bialgebras were introduced by Joni and Rota [1 ...
... An infinitesimal bialgebra (abbreviated ǫ-bialgebra) is a triple (A, µ, ∆) where (A, µ) is an associative algebra, (A, ∆) is a coassociative coalgebra, and ∆ is a derivation (see Section 2). We write ∆(a) = a1⊗a2 , omitting the sum symbol. Infinitesimal bialgebras were introduced by Joni and Rota [1 ...
full text (.pdf)
... of B, and the choice operator + acts as disjunction. Intuitively, a test bc succeeds i both b and c succeed, and b + c succeeds i either b or c succeeds. Since b 1 for all b 2 B, it is tempting to de ne tests in an arbitrary Kleene algebra K to be the set fp 2 K j p 1g. Although this makes sen ...
... of B, and the choice operator + acts as disjunction. Intuitively, a test bc succeeds i both b and c succeed, and b + c succeeds i either b or c succeeds. Since b 1 for all b 2 B, it is tempting to de ne tests in an arbitrary Kleene algebra K to be the set fp 2 K j p 1g. Although this makes sen ...
Problems 3.6 - Number Theory Web
... (iii) S is closed under scalar multiplication. For let [x, y] ∈ S and t ∈ R. Then x = 2y and hence tx = 2(ty). Consequently [tx, ty] = t[x, y] ∈ S. (b) Let S be the set of vectors [x, y] satisfying x = 2y and 2x = y. Then S is a subspace of R2 . This can be proved in the same way as (a), or alternat ...
... (iii) S is closed under scalar multiplication. For let [x, y] ∈ S and t ∈ R. Then x = 2y and hence tx = 2(ty). Consequently [tx, ty] = t[x, y] ∈ S. (b) Let S be the set of vectors [x, y] satisfying x = 2y and 2x = y. Then S is a subspace of R2 . This can be proved in the same way as (a), or alternat ...
Linear models 2
... using the fact that for diagonal matrices applying the matrix repeatedly is equivalent to taking the power of the diagonal entries. This allows to compute the k matrix products using just 3 matrix products and taking the power of n numbers. From high-school or undergraduate algebra you probably reme ...
... using the fact that for diagonal matrices applying the matrix repeatedly is equivalent to taking the power of the diagonal entries. This allows to compute the k matrix products using just 3 matrix products and taking the power of n numbers. From high-school or undergraduate algebra you probably reme ...
On the Extension of Complex Numbers - Rose
... through the angle arg(z) about the origin and a re-scaling of the modulus by |z|. Of course, the rôles of z and w could be exchanged in the previous statement, reflecting commutativity under multiplication of ordinary complex numbers. In defining multiplication for supercomplex numbers we shall pur ...
... through the angle arg(z) about the origin and a re-scaling of the modulus by |z|. Of course, the rôles of z and w could be exchanged in the previous statement, reflecting commutativity under multiplication of ordinary complex numbers. In defining multiplication for supercomplex numbers we shall pur ...
Fall 2007 Exam 2
... Note that even though A has a row of zeros, AT A does not have a row of zeros. Moreover, A is a 4 × 3 matrix, so det A is not defined. (b) (3 points) Your friend (who, sadly, is not enrolled in Linear Algebra) claims that there is no such thing as 4-space, and thus, there is no such thing as a 3-box ...
... Note that even though A has a row of zeros, AT A does not have a row of zeros. Moreover, A is a 4 × 3 matrix, so det A is not defined. (b) (3 points) Your friend (who, sadly, is not enrolled in Linear Algebra) claims that there is no such thing as 4-space, and thus, there is no such thing as a 3-box ...
Definitions of Linear Algebra Terms
... There will be a question on each of the exams that asks you to state some de…nitions. It will be required that you state these de…nitions very precisely in order to receive credit for the exam question. (Seemingly very small things such as writing the word “or” instead of the word “and”, for example ...
... There will be a question on each of the exams that asks you to state some de…nitions. It will be required that you state these de…nitions very precisely in order to receive credit for the exam question. (Seemingly very small things such as writing the word “or” instead of the word “and”, for example ...
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs. The exterior product of two vectors u and v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = −(v ∧ u) for all vectors u and v. One way to visualize a bivector is as a family of parallelograms all lying in the same plane, having the same area, and with the same orientation of their boundaries—a choice of clockwise or counterclockwise.When regarded in this manner, the exterior product of two vectors is called a 2-blade. More generally, the exterior product of any number k of vectors can be defined and is sometimes called a k-blade. It lives in a space known as the kth exterior power. The magnitude of the resulting k-blade is the volume of the k-dimensional parallelotope whose edges are the given vectors, just as the magnitude of the scalar triple product of vectors in three dimensions gives the volume of the parallelepiped generated by those vectors.The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, blades have a concrete geometric interpretation, and objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms, and the pairing between the exterior algebra and its dual is given by the interior product.