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This paper outlines our ideas on how to teach linear algebra in a mechanized mathematical environment, and discusses some of our reasons for thinking that this is a better way to teach linear algebra than the ``old fashioned way''. We…
So far, the scope of computer algebra has been needlessly restricted to exact algebraic methods. Its possible extension to approximate analytical methods is discussed. The entangled roles of functional analysis and symbolic programming,…
In the setting of modern mathematical logic and model theory, classification theory has been one of the landmark achievements of the field. Likewise, the classification of UHF-algebras and AF-algebras were substantial contributions to the…
We consider algorithms with access to an unknown matrix $M\in\mathbb{F}^{n \times d}$ via matrix-vector products, namely, the algorithm chooses vectors $\mathbf{v}^1, \ldots, \mathbf{v}^q$, and observes $M\mathbf{v}^1,\ldots,…
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms…
Efficient numerical linear algebra is a core ingredient in many applications across almost all scientific and industrial disciplines. With this survey we want to illustrate that numerical linear algebra has played and is playing a crucial…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
As dynamic and control systems become more complex, relying purely on numerical computations for systems analysis and design might become extremely expensive or totally infeasible. Computer algebra can act as an enabler for analysis and…
This paper presents an algebraic theory of linear signal processing. At the core of algebraic signal processing is the concept of a linear signal model defined as a triple (A, M, phi), where familiar concepts like the filter space and the…
Transformers can learn to perform numerical computations from examples only. I study nine problems of linear algebra, from basic matrix operations to eigenvalue decomposition and inversion, and introduce and discuss four encoding schemes to…
We present a formal language with expressions denoting general symbol structures and queries which access information in those structures. A sequence-to-sequence network processing this language learns to encode symbol structures and query…
This work gives an overview of analytic tools for the design, analysis, and modelling of communication systems which can be described by linear vector channels such as y = Hx+z where the number of components in each vector is large. Tools…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
In finance, economics and many other fields, observations in a matrix form are often observed over time. For example, many economic indicators are obtained in different countries over time. Various financial characteristics of many…
In this book, there are five chapters: Systems of Linear Equations, Vector Spaces, Homogeneous Systems, Characteristic Equation of Matrix, and Matrix Dot Product. It has also exercises at the end of each chapter above to let students…
We investigate some types of composition operators, linear and not, and conditions for some spaces to be mapped into themselves and for the operators to satisfy some good properties.
In this article, a novel approach to learning a complex function which can be written as the system of linear equations is introduced. This learning is grounded upon the observation that solving the system of linear equations by a…
Based on the property that solving the system of linear matrix equations via the column space and the row space projections boils down to an approximation in the least squares error sense, a formulation for learning the weight matrices of…
This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…