Vector Algebra
... After studying this chapter, you should be able to understand • Addition, subtraction, scalar product and vector product and solve problems based on the above. 191. VECTORS ...
... After studying this chapter, you should be able to understand • Addition, subtraction, scalar product and vector product and solve problems based on the above. 191. VECTORS ...
1 Basis
... then any solution, since it is an element of N ull(A), is a linear combination x = α1 v1 + ... + αn−r vn−r . Now turn to non-homogenous systems. Let A · x = b, x ∈ Rn , b ∈ Rm be a system of linear equations and A · x = 0 be the corresponding homogenous system. Theorem 13 Let c be a particular solut ...
... then any solution, since it is an element of N ull(A), is a linear combination x = α1 v1 + ... + αn−r vn−r . Now turn to non-homogenous systems. Let A · x = b, x ∈ Rn , b ∈ Rm be a system of linear equations and A · x = 0 be the corresponding homogenous system. Theorem 13 Let c be a particular solut ...
Boolean Algebra
... Given a truth table for a Boolean function, construction of the product-of-sums representation is trivial: - for each row in which the function value is 0, form a product term involving all the variables, taking the variable if its value is 1 and the complement if the variable's value is 0 - take th ...
... Given a truth table for a Boolean function, construction of the product-of-sums representation is trivial: - for each row in which the function value is 0, form a product term involving all the variables, taking the variable if its value is 1 and the complement if the variable's value is 0 - take th ...
Vector Spaces, Affine Spaces, and Metric Spaces
... If {u1 , . . . , um } and {v1 , . . . , vn } both satisfy these conditions then m = n. Definition 2.6 A finite set {v1 , . . . , vn } ⊆ V of a vector space is called a basis if it satisfies one, and hence all, of the conditions in Theorem 2.1. The unique number of elements in a basis is called the d ...
... If {u1 , . . . , um } and {v1 , . . . , vn } both satisfy these conditions then m = n. Definition 2.6 A finite set {v1 , . . . , vn } ⊆ V of a vector space is called a basis if it satisfies one, and hence all, of the conditions in Theorem 2.1. The unique number of elements in a basis is called the d ...
full text (.pdf)
... combines Kleene algebra (KA), or the algebra of regular expressions, with Boolean algebra. One can model basic programming language constructs such as conditionals and while loops, verification conditions, and partial correctness assertions. KAT has been applied successfully in verification tasks in ...
... combines Kleene algebra (KA), or the algebra of regular expressions, with Boolean algebra. One can model basic programming language constructs such as conditionals and while loops, verification conditions, and partial correctness assertions. KAT has been applied successfully in verification tasks in ...
Linear spaces and linear maps Linear algebra is about linear
... Ex: i) The space of all m by n matrices forms a vector space Mat(m,n) where A+B is the matrix whose (i,j) entry, i.e. the entry in the ith row and jth column, is the sum of the (i,j) entries of A and B, and where cA is the matrix whose (i,j) entry is c times the (i,j) entry of A. ii) The space Hom(k ...
... Ex: i) The space of all m by n matrices forms a vector space Mat(m,n) where A+B is the matrix whose (i,j) entry, i.e. the entry in the ith row and jth column, is the sum of the (i,j) entries of A and B, and where cA is the matrix whose (i,j) entry is c times the (i,j) entry of A. ii) The space Hom(k ...
Equivariant Cohomology
... HomC pX, Xqrηs with degpηq “ ´1 and dpηq “ idX . One can use this construction to localize categories: we quotient by the full subcategory whose objects consist of mapping cones of morphisms we wish to invert. Example 5. In light of the above remark, we can write, with degpβq “ 2 and degpλq “ ´1, H‚ ...
... HomC pX, Xqrηs with degpηq “ ´1 and dpηq “ idX . One can use this construction to localize categories: we quotient by the full subcategory whose objects consist of mapping cones of morphisms we wish to invert. Example 5. In light of the above remark, we can write, with degpβq “ 2 and degpλq “ ´1, H‚ ...
A primer of Hopf algebras
... 1.1. After the pioneer work of Connes and Kreimer1 , Hopf algebras have become an established tool in perturbative quantum field theory. The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940’s dealing with the cohomology of compact Lie groups and their homogeneous sp ...
... 1.1. After the pioneer work of Connes and Kreimer1 , Hopf algebras have become an established tool in perturbative quantum field theory. The notion of Hopf algebra emerged slowly from the work of the topologists in the 1940’s dealing with the cohomology of compact Lie groups and their homogeneous sp ...
Holt McDougal Algebra 1
... represent a value that can change. (ie. X, y, a, n) A constant is a value that does not change. (ie. 5, -7, or 356) A numerical expression contains only constants and operations. (ie. 5+7-12*35.3) An algebraic expression may contain variables, constants, and operations. (5x-7a+3) To evaluate an expr ...
... represent a value that can change. (ie. X, y, a, n) A constant is a value that does not change. (ie. 5, -7, or 356) A numerical expression contains only constants and operations. (ie. 5+7-12*35.3) An algebraic expression may contain variables, constants, and operations. (5x-7a+3) To evaluate an expr ...
Boolean algebra
... In addition, certain axioms must be satisfied: - closure properties for both binary operations and the unary operation - associativity of each binary operation over the other, - commutativity of each each binary operation, - distributivity of each binary operation over the other, ...
... In addition, certain axioms must be satisfied: - closure properties for both binary operations and the unary operation - associativity of each binary operation over the other, - commutativity of each each binary operation, - distributivity of each binary operation over the other, ...
CONVERGENCE THEOREMS FOR PSEUDO
... Let E := (E, τ ) be a locally convex algebra in which each element is bounded (i.e., E = E0 , in the notation above). Let B = {B} denote a family of bounded, absolutely convex, closed subsets B of E such that B 2 ⊂ B and that each such subset of B is contained in some B in B. By Propositions (2.2)-( ...
... Let E := (E, τ ) be a locally convex algebra in which each element is bounded (i.e., E = E0 , in the notation above). Let B = {B} denote a family of bounded, absolutely convex, closed subsets B of E such that B 2 ⊂ B and that each such subset of B is contained in some B in B. By Propositions (2.2)-( ...
Banach Algebra Notes - Oregon State Mathematics
... (a) a set C ⊆ X is closed if and only if whenever {xα } is a net in C with limit x in X then we have that x is in C, and (b) a function f : X → Y is continuous if and only if xα → x implies f (xα ) → f (x) for all nets. In the proof of both parts, we consider the directed set Tx , the set of neighbo ...
... (a) a set C ⊆ X is closed if and only if whenever {xα } is a net in C with limit x in X then we have that x is in C, and (b) a function f : X → Y is continuous if and only if xα → x implies f (xα ) → f (x) for all nets. In the proof of both parts, we consider the directed set Tx , the set of neighbo ...
Linear Span and Bases 1 Linear span
... be n, then to check that a list of n vectors is a basis it is enough to check whether it spans V (resp. is linearly independent). Proof. To prove point 1, let (u1, . . . , um ) be a basis of U. This list is linearly independent both in U and V . By the Basis Extension Theorem 7 it can be extended to ...
... be n, then to check that a list of n vectors is a basis it is enough to check whether it spans V (resp. is linearly independent). Proof. To prove point 1, let (u1, . . . , um ) be a basis of U. This list is linearly independent both in U and V . By the Basis Extension Theorem 7 it can be extended to ...
Existence of almost Cohen-Macaulay algebras implies the existence
... A big Cohen-Macaulay algebra over a local ring (R, m) is an R-algebra B such that some system of parameters of R is a regular sequence on B. It is balanced if every system of parameters of R is a regular sequence on B. Big Cohen-Macaulay algebras exist in equal characteristic [7], [6] and also in mi ...
... A big Cohen-Macaulay algebra over a local ring (R, m) is an R-algebra B such that some system of parameters of R is a regular sequence on B. It is balanced if every system of parameters of R is a regular sequence on B. Big Cohen-Macaulay algebras exist in equal characteristic [7], [6] and also in mi ...
Notes: Orthogonal transformations and isometries
... general sense: A subspace of R3 can be a completely arbitrary subset, with the metric (or if you know topology, the subspace topology) that it inherits from the usual metric on R3 . Thus a vector subspace of R3 , or more generally of Rn , is also a subspace in this topological sense, but not conver ...
... general sense: A subspace of R3 can be a completely arbitrary subset, with the metric (or if you know topology, the subspace topology) that it inherits from the usual metric on R3 . Thus a vector subspace of R3 , or more generally of Rn , is also a subspace in this topological sense, but not conver ...
MATH 304 Linear Algebra Lecture 13: Span. Spanning
... • upper triangular matrices: c = 0 • lower triangular matrices: b = 0 • symmetric matrices (AT = A): b = c • anti-symmetric matrices (AT = −A): a = d = 0 and c = −b • matrices with zero trace: a + d = 0 (trace = the sum of diagonal entries) ...
... • upper triangular matrices: c = 0 • lower triangular matrices: b = 0 • symmetric matrices (AT = A): b = c • anti-symmetric matrices (AT = −A): a = d = 0 and c = −b • matrices with zero trace: a + d = 0 (trace = the sum of diagonal entries) ...
Solvable Affine Term Structure Models
... fields f (V) and when ΦR exists, it maps a non linear ODE into a linear one. It is well known (see e.g. Walcher 1991, Proposition 8.7) that the existence of such a change of coordinates that linearizes the ODE implies the existence of a finite dimensional Lie subalgebra containing L, and this notion ...
... fields f (V) and when ΦR exists, it maps a non linear ODE into a linear one. It is well known (see e.g. Walcher 1991, Proposition 8.7) that the existence of such a change of coordinates that linearizes the ODE implies the existence of a finite dimensional Lie subalgebra containing L, and this notion ...
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.