The almost split triangles for perfect complexes over gentle algebras

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

It is a standard and very useful technique to approximate geometric objects and maps by linear objects and maps. A principal (trivial) example in basic analysis is approximating an open subspace U of \(\mathbb{K}^{n}\) at a point by the linear space \(\mathbb{K}^{n}\) itself by visualizing \(\mathbb{K}^{n}\) as a tangent space to U at the given point. Another (less trivial) example is the derivative of a differentiable map f on U in a point that is the best approximation of f nearby this point by a linear map. In the first section we extend this technique to premanifolds and morphisms of premanifolds. Thus we first define the tangent space of a premanifold at a point, which will be a \(\mathbb{K}\)-vector space that we visualize as the ”best linear approximation“ of the premanifold at the given point. Then we define the derivative in a point x of a morphism f. This will be a linear map from tangent space at x to the tangent space at \(f(x)\).

In this closing chapter we will be engaged again with notions and techniques typical for the field of Linear Algebra, but here our attention will also be focused on the abstract structure of vector spaces and linear mappings (or linear transformations) between them. Vectors—the elements of a vector space—do not need to be ordered lists of numbers, polynomials can also be viewed as vectors. Of course, matrices continue to play a prominent role, namely as representations of linear mappings between vector spaces with predetermined bases.KeywordsVector SpaceSingular Value DecompositionMatrix RepresentationCharacteristic PolynomialNull SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Linear Algebra Linear algebra studies linear transformations and vector spaces, or in another language, matrix multiplication and the vector space R n . You should know how to translate between the language of abstract vector spaces and the language of matrices. In particular, given a basis for a vector space, you should know how to represent any linear transformation as a matrix. Further, given two matrices, you should know how to determine if these matrices actually represent the same linear transformation, but under different choices of bases. The key theorem of linear algebra is a statement that gives many equivalent descriptions for when a matrix is invertible. These equivalences should be known cold. You should also know why eigenvectors and eigenvalues occur naturally in linear algebra. Real Analysis The basic definitions of a limit, continuity, differentiation and integration should be known and understood in terms of ∈'s and δ's. Using this ∈ and δ language, you should be comfortable with the idea of uniform convergence of functions. Differentiating Vector-Valued Functions The goal of the Inverse Function Theorem is to show that a differentiable function f : R n → R n is locally invertible if and only if the determinant of its derivative (the Jacobian) is non-zero. You should be comfortable with what it means for a vector-valued function to be differentiable, why its derivative must be a linear map (and hence representable as a matrix, the Jacobian) and how to compute the Jacobian.

Geometrically, curves and surfaces are usually considered to be sets of points with some special properties, living in a space consisting of “points.” Typically, one is also interested in geometric properties invariant under certain transformations, for example, translations, rotations, projections, etc. One could model the space of points as a vector space, but this is not very satisfactory for a number of reasons. One reason is that the point corresponding to the zero vector (0), called the origin, plays a special role, when there is really no reason to have a privileged origin. Another reason is that certain notions, such as parallelism, are handled in an awkward manner. But the deeper reason is that vector spaces and affine spaces really have different geometries. The geometric properties of a vector space are invariant under the group of bijective linear maps, whereas the geometric properties of an affine space are invariant under the group of bijective affine maps, and these two groups are not isomorphic. Roughly speaking, there are more affine maps than linear maps.

Based on a completely distributive lattice L, the degree to which an L-subset in a vector space is an L-fuzzy subspace is introduced via the implication operation on L. By using four kinds of cut sets, several characterizations of L-fuzzy subspace degree are given. Further, it is shown that the L-fuzzy subspace degree of a vector space can induce an L-fuzzy convexity in a natural way and the linear mappings between vector spaces are L-fuzzy convexity-preserving and L-fuzzy convex-to-convex mappings between the induced L-fuzzy convex spaces.

Amapping from one vector space to another vector space over the same field is linear if it is additive and it preserves scalar multiplication. This note demonstrates that a linear mapping is, in fact, a particular example from a class of mappings known as semilinear mappings. In the case of complex vector space a semilinear mapping may be linear or conjugate linear and, unlike the real case, a bounded additive mapping is the sum of a linear and a conjugate linear mapping. The history and relevance of the semilinear mappings in mathematical physics and in pure mathematics are indicated.

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, and then instantiate the abstract setting to sets and relations and to finite-dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.

On r-potent linear mappings

Let E be a vector space, F aset, G be a locally convex space, b : E X F → G a map such that b(·,y): E → G is linear for every y ∈ F; we write b(x, y) = x · y for brevity. Let λ be a scalar sequence space and w(E,F) the weakest topology on E such that the linear maps b(·,y): E → G are continuous for all y ∈ F .A series ΣjXj in X is λ multiplier convergent with respect to w(E, F) if for each t = {tj} ∈ λ ,the series Σj=1∞ tjxj is w(E,F) convergent in E. For multiplier spaces λ satisfying certain gliding hump properties we establish the following uniform convergence result: Suppose Σjxij is λ multiplier convergent with respect to w(E, F) for each i ∈ N and for each t ∈ λ the set {Σj=1∞ tjxij : i} is uniformly bounded on any subset B C F such that {x · y : y ∈ B} is bounded for x ∈ E.Then for each t ∈ λ the series Σj=1∞ tjxij · y converge uniformly for y ∈ B,i ∈ N. This result is used to prove a Hahn-Schur Theorem for series such that limi Σj=1∞ tjxij · y exists for t ∈ λ,y ∈ F. Applications of these abstract results are given to spaces of linear operators, vector spaces in duality, spaces of continuous functions and spaces with Schauder bases.

Suppose q ≥ 2 and I is a field with at least q elements. Let Ar ,= 1,….be nonzero linear maps (with appropriate vector spaces over F as domains and codomains), at most one of rank one. and let Br .i = 1…,q, be linear maps so that At and Bt have the same domain and codomain. Then each of (i)B1x⊗ ⋯ ⊗ Bqx = A1x ⊗ ⋯ ⊗ Aqxfor all x; (ii)B1x1 v ⋯ v Bqxq = A1x1 v ⋯ v Aqxq for all x1… xq and (iii)B1xv ⋯ v Bqx = A1x v ⋯ v Aqx for all x; is sufficient as well as necessary for the natural corresponding condition of linear dependence of the maps At and Bt .

This methodological study provides a step-by-step introduction to a computational implementation of word and paradigm morphology using linear mappings between vector spaces for form and meaning. Taking as starting point the linear regression model, the main concepts underlying linear mappings are introduced and illustrated with R code. It is then shown how vector spaces can be set up for Latin verb conjugations, using 672 inflected variants of two verbs each from the four main conjugation classes. It turns out that mappings from form to meaning (comprehension), and from meaning to form (production) can be carried out loss-free. This study concludes with a demonstration that when the graph of triphones, the units that underlie the form space, is mapped onto a 2-dimensional space with a self-organising algorithm from physics (graphopt), morphological functions show topological clustering, even though morphemic units do not play any role whatsoever in the model. It follows, first, that evidence for morphemes emerging from experimental studies using, for instance, fMRI, to localize morphemes in the brain, does not guarantee the existence of morphemes in the brain, and second, that potential topological organization of morphological form in the cortex may depend to a high degree on the morphological system of a language.

Data represented geometrically in high-dimensional vector spaces can be found in many applications. Images and videos, are often represented by assigning a dimension for every pixel (and time). Text documents may be represented in a vector space where each word in the dictionary incurs a dimension. The need to manipulate such data in huge corpora such as the web and to support various query types gives rise to the question of how to represent the data in a lower-dimensional space to allow more space and time efficient computation. Linear mappings are an attractive approach to this problem because the mapped input can be readily fed into popular algorithms that operate on linear spaces (such as principal-component analysis, PCA) while avoiding the curse of dimensionality. The fact that such mappings even exist became known in computer science following seminal work by Johnson and Lindenstrauss in the early 1980s. The underlying technique is often called "random projection." The complexity of the mapping itself, essentially the product of a vector with a dense matrix, did not attract much attention until recently. In 2006, we discovered a way to "sparsify" the matrix via a computational version of Heisenberg's Uncertainty Principle. This led to a significant speedup, which also retained the practical simplicity of the standard Johnson-Lindenstrauss projection. We describe the improvement in this article, together with some of its applications.

Linear Algebra offers a unified treatment of both matrix-oriented and theoretical approaches to the course, which will be useful for classes with a mix of mathematics, physics, engineering, and computer science students. Major topics include singular value decomposition, the spectral theorem, linear systems of equations, vector spaces, linear maps, matrices, eigenvalues and eigenvectors, linear independence, bases, coordinates, dimension, matrix factorizations, inner products, norms, and determinants.

Given an matrix A, considered as a linear map A : ℝn⟶ℝn, then A induces a topological space structure on ℝn which differs quite a lot from the usual one (induced by the Euclidean metric). This new topological structure on ℝn has very interesting properties with a nice special geometric flavor, and it is a particular case of the so called “primal space,” In particular, some algebraic information can be shown in a topological fashion and the other way around. If X is a non‐empty set and f : X⟶X is a map, there exists a topology τf induced on X by f, defined by τf = {U ⊂ X : f−1(U) ⊂ U}. The pair (X, τf) is called the primal space induced by f. In this paper, we investigate some characteristics of primal space structure induced on the vector space ℝn by matrices; in particular, we describe geometrical properties of the respective spaces for the case.