these notes - MIT Mathematics - Massachusetts Institute of Technology
... isomorphism of vector spaces Γ(X, L ) → Γ(Y, M ). 2. The group G(L ) Now let X be an abelian variety over k. Given x ∈ X(k), let τx : X → X be translation by x. Then one can define G(L ) as the group of automorphisms (f, t) of L → X such that t is a translation τx for some x ∈ X(k). (Given x, for su ...
... isomorphism of vector spaces Γ(X, L ) → Γ(Y, M ). 2. The group G(L ) Now let X be an abelian variety over k. Given x ∈ X(k), let τx : X → X be translation by x. Then one can define G(L ) as the group of automorphisms (f, t) of L → X such that t is a translation τx for some x ∈ X(k). (Given x, for su ...
Sheet 14 - TCD Maths home
... Now, for any choice of a, b, c, we can find a solution (k1 , k2 , k3 , k4 ). Hence any v can be written as a linear combination k1 v1 + k2 v2 + k3 v3 + k4 v4 and therefore the vectors span IR3 . ...
... Now, for any choice of a, b, c, we can find a solution (k1 , k2 , k3 , k4 ). Hence any v can be written as a linear combination k1 v1 + k2 v2 + k3 v3 + k4 v4 and therefore the vectors span IR3 . ...
A NOTE ON NORMAL VARIETIES OF MONOUNARY ALGEBRAS 1
... 3. Normal varieties of monounary algebras A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. For every variety V let N (V ) denote the smallest normal variety (of the same type as V ) containing V . Remark. From the results in Sect ...
... 3. Normal varieties of monounary algebras A variety is called normal if no laws of the form s = t are valid in it where s is a variable and t is not a variable. For every variety V let N (V ) denote the smallest normal variety (of the same type as V ) containing V . Remark. From the results in Sect ...
The Zero-Sum Tensor
... Department of Game Design Uppsala University Visby, Sweden [email protected] Abstract—The zero-sum matrix, or in general, tensor, reveals some consistent properties at multiplication. In this paper, three mathematical rules are derived for multiplication involving such entities. The ...
... Department of Game Design Uppsala University Visby, Sweden [email protected] Abstract—The zero-sum matrix, or in general, tensor, reveals some consistent properties at multiplication. In this paper, three mathematical rules are derived for multiplication involving such entities. The ...
Sheet 9
... Thus every row of the new matrix is a multiple of the vector [y1 , y2 , . . . , yn ] and every column a multiple of the vector [ x1 , x2 , . . . , xm ]. • Consider a website for film reviews. In a simplified version, users can vote +1 if they like a movie, -1 if they do not like it and 0 if they are ...
... Thus every row of the new matrix is a multiple of the vector [y1 , y2 , . . . , yn ] and every column a multiple of the vector [ x1 , x2 , . . . , xm ]. • Consider a website for film reviews. In a simplified version, users can vote +1 if they like a movie, -1 if they do not like it and 0 if they are ...
01 Introduction.pdf
... • Linear algebra provides mathematical tools – Vectors, coordinate systems, matrices, etc. – As little abstract theory as possible in this ...
... • Linear algebra provides mathematical tools – Vectors, coordinate systems, matrices, etc. – As little abstract theory as possible in this ...
Math 261y: von Neumann Algebras (Lecture 14)
... The converse is true as well: any distributive lattice in which every element x has a complement x0 can be regarded as a Boolean algebra, with multiplication given by xy = x ∧ y and addition given by x ⊕ y = (x ∨ y) ∧ (x0 ∨ y 0 ). If B is a Boolean algebra, we will say that a pair of elements x, y ∈ ...
... The converse is true as well: any distributive lattice in which every element x has a complement x0 can be regarded as a Boolean algebra, with multiplication given by xy = x ∧ y and addition given by x ⊕ y = (x ∨ y) ∧ (x0 ∨ y 0 ). If B is a Boolean algebra, we will say that a pair of elements x, y ∈ ...
Week_1_LinearAlgebra..
... • If the spanning set is linearly independent, it’s also known as a basis for that subspace • The coordinate representation of a vector in a subspace is unique with respect to a basis for that subspace ...
... • If the spanning set is linearly independent, it’s also known as a basis for that subspace • The coordinate representation of a vector in a subspace is unique with respect to a basis for that subspace ...
Slide 1
... • „fine grid“, „coarse grid“ only makes sense if the problem to solve corresponds to a grid – this is the case in the finite difference methods described before – need to find an “interpolation matrix” for the propagation step to generate the coarser grid (for instance simple linear interpolatio ...
... • „fine grid“, „coarse grid“ only makes sense if the problem to solve corresponds to a grid – this is the case in the finite difference methods described before – need to find an “interpolation matrix” for the propagation step to generate the coarser grid (for instance simple linear interpolatio ...
Section 3.3 Vectors in the Plane
... Now, if v is any vector that makes an angle θ with the positive x-axis, it has the same direction as u and v= ...
... Now, if v is any vector that makes an angle θ with the positive x-axis, it has the same direction as u and v= ...
Sample Problems for Midterm 2 1 True or False: 1.1 If V is a vector
... 1.13 If dim V = n, then any independent set in V has at least n elements. 1.14 If dim V = n, then any independent set in V with n elements is a basis for V. 1.15 If W is a subspace of V, then dim W ≤ dim V. 1.16 If W = spn S and S is linearly independent, then S is a basis for W. 1.17 [v − w] S = [ ...
... 1.13 If dim V = n, then any independent set in V has at least n elements. 1.14 If dim V = n, then any independent set in V with n elements is a basis for V. 1.15 If W is a subspace of V, then dim W ≤ dim V. 1.16 If W = spn S and S is linearly independent, then S is a basis for W. 1.17 [v − w] S = [ ...
Review for Exam 2 Solutions Note: All vector spaces are real vector
... 4. Let U and W be subspaces of a vector space V . Let U + W be the set of all vectors in V that have the form u + w for some u in U and w in W . (a) Show that U + W is a subspace of V . The set U + W is nonempty - in fact it contains both U and W since both spaces contain 0. To check if U + W is clo ...
... 4. Let U and W be subspaces of a vector space V . Let U + W be the set of all vectors in V that have the form u + w for some u in U and w in W . (a) Show that U + W is a subspace of V . The set U + W is nonempty - in fact it contains both U and W since both spaces contain 0. To check if U + W is clo ...
Reed-Muller codes
... they are not perfect. First, we cover the necessary material on the Boolean algebra. We then define the Reed-Muller codes R(r, m) as certain subspaces of the Boolean algebra on V m . We calculate the parameters of R(r, m). In the exercises to this chapter, some of the previously known codes are real ...
... they are not perfect. First, we cover the necessary material on the Boolean algebra. We then define the Reed-Muller codes R(r, m) as certain subspaces of the Boolean algebra on V m . We calculate the parameters of R(r, m). In the exercises to this chapter, some of the previously known codes are real ...
Cohomology of Lie groups and Lie algebras
... The aim of this expository essay is to illustrate one example of a local-to-global phenomenon. What we mean by that is better illustrated by explaining the topic at hand. We aim to understand the de Rham cohomology groups of a Lie group. But instead of doing it using actual differential forms, we sh ...
... The aim of this expository essay is to illustrate one example of a local-to-global phenomenon. What we mean by that is better illustrated by explaining the topic at hand. We aim to understand the de Rham cohomology groups of a Lie group. But instead of doing it using actual differential forms, we sh ...
Vector Spaces
... If c is a scalar (a real number), then the scalar multiple of the vector x by the scalar c, denoted by cx, is the vector cx (cx1 , cx2 ,..., cxn ) Note: (-1)x = -x = ( x1 , x2 ,..., xn ) Vector Space: Let V be a set vectors in which the operations of sum of vectors and of scalar multiplication ...
... If c is a scalar (a real number), then the scalar multiple of the vector x by the scalar c, denoted by cx, is the vector cx (cx1 , cx2 ,..., cxn ) Note: (-1)x = -x = ( x1 , x2 ,..., xn ) Vector Space: Let V be a set vectors in which the operations of sum of vectors and of scalar multiplication ...
Semisimple algebras and Wedderburn`s theorem
... to the direct sums of matrix algebras in question. As we now know the algebra A explicitly, it will not be hard to identify the simple A-modules. We shall write elements of A as x = (x1 , . . . , xl ) where xi ∈ Mmi (C). For i = 1, . . . , l, we may define the structure of an A-module on the vector ...
... to the direct sums of matrix algebras in question. As we now know the algebra A explicitly, it will not be hard to identify the simple A-modules. We shall write elements of A as x = (x1 , . . . , xl ) where xi ∈ Mmi (C). For i = 1, . . . , l, we may define the structure of an A-module on the vector ...
1. Topology Here are some basic definitions concerning topological
... representation of a matrix Lie group G we obtain a Lie algebra homomorphism dπ : g 7→ gl(n, C) (u(n)). In this case, dπ(X) is called the infinitesimal generator corresponding to X. Remark 4.15. It is of course not at all clear a priori why a Lie group homomorphism φ, which to begin with only is assu ...
... representation of a matrix Lie group G we obtain a Lie algebra homomorphism dπ : g 7→ gl(n, C) (u(n)). In this case, dπ(X) is called the infinitesimal generator corresponding to X. Remark 4.15. It is of course not at all clear a priori why a Lie group homomorphism φ, which to begin with only is assu ...
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.