On a generalization of linear independence in finite-dimensional vector spaces

Summary. In this paper, we discuss the properties that hold in finite dimensional vector spaces and related spaces. In the Mizar language [1], [2], variables are strictly typed, and their type conversion requires a complicated process. Our purpose is to formalize that some properties of finite dimensional vector spaces are preserved in type transformations, and to contain the complexity of type transformations into this paper. Specifically, we show that properties such as algebraic structure, subsets, finite sequences and their sums, linear combination, linear independence, and affine independence are preserved in type conversions among TOP-REAL(n), REAL-NS(n), and n-VectSp over F Real. We referred to [4], [9], and [8] in the formalization.

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A closer look at the stable completion

10.23943/princeton/9780691161686.003.0005 ◽

2017 ◽

Author(s):

Ehud Hrushovski ◽

François Loeser

Keyword(s):

Inverse Limit ◽

Unit Vector ◽

Vector Spaces ◽

Dimensional Vector ◽

Compact Sets ◽

Linear Topology ◽

Topological Embedding ◽

Definable Set ◽

Finite Dimensional ◽

The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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Orbit-Finite-Dimensional Vector Spaces and Weighted Register Automata

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) ◽

10.1109/lics52264.2021.9470634 ◽

2021 ◽

Author(s):

Mikolaj Bojanczyk ◽

Bartek Klin ◽

Joshua Moerman

Keyword(s):

Vector Spaces ◽

Dimensional Vector ◽

Finite Dimensional

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A note on extensions of multilinear maps defined on multilinear varieties

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000055 ◽

2021 ◽

pp. 1-26

Author(s):

W. T. Gowers ◽

L. Milićević

Keyword(s):

Finite Fields ◽

Vector Spaces ◽

Restriction Map ◽

Zero Set ◽

Dimensional Vector ◽

Prime Field ◽

Finite Dimensional ◽

Multilinear Maps ◽

Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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