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On the one hand the algebras of linear operators here act on finite-dimensional vector spaces, and on the other hand the point of view is generally an analysts'. Also, one might think of algebras as being used to add more data to basic…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
There is strong interest in developing mathematical methods that can be used to understand complex neural networks used in image analysis. In this paper, we introduce techniques from Linear Algebra to model neural network layers as maps…
We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ over a fixed field $\mathbb{F}$ for a specified…
An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative field is given. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.…
Linear systems often involve, as a basic building block, solutions of equations of the form \begin{align*} A_Sx_S&+A_Px_P =0\\ A'_Sx_S & =0, \end{align*} where our primary interest might be in the vector variable $x_P.$ Usually, neither…
Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra and computational algebra), geometry and combinatorics to provide insight into knotty problems in mathematical statistics. In this survey…
Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…
The division between two vectors belonging to the same vector space is obtained by elementary procedures of vector algebra and is defined by a matrix. This representation is obtained for two and three dimensional vector spaces. A new vector…
While there has been some discussion on how Symbolic Computation could be used for AI there is little literature on applications in the other direction. However, recent results for quantifier elimination suggest that, given enough example…
There are versions of "calculus" in many settings, with various mixtures of algebra and analysis. In these informal notes we consider a few examples that suggest a lot of interesting questions.
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
Linear systems under the influence of nonlinear and random linear perturbations, and with random initial and boundary conditions, are discussed. The notion of states of a system is substituted by the notion of the generating vectors for…
We usually define an algebraic structure by a set, some operations defined on this set and some propositions that the algebraic structure must validate. In some cases, we can replace these propositions by an algorithm on terms constructed…
We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex…
In this paper, some real-world motivated examples are provided illustrating the power of linear algebra tools as the product of matrices, determinants, eigenvalues and eigenvectors. In this sense, some practical applications related to…
Vector calculus in three-dimensional space is ubiquitous in applications of mathematics in physics and engineering. Its two-dimensional version is, however, quite rare. Here we try to provide a pedagogical account of the subject. It is…
Large language models (LLMs) were invented for natural language tasks such as translation, but they have proved that they can perform highly complex functions across domains. Additionally, they have been thought to develop new skills…
Thye theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, sampling theory and much more, as well as being a fruitful area of research in abstract mathematics. In this…
In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the…