Linear independence and dependence

1978 ◽  
pp. 33-39
Author(s):  
Larry Smith
Keyword(s):  
Linear Independence

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

scholarly journals Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

2021 ◽  
Author(s):  
Florian Mannel
Keyword(s):  
Linear Independence ◽  
Affine Subspace ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Convergence Properties ◽  
Initial Matrix ◽  
First Time ◽  
Special Case ◽  
Dimensional Mapping ◽  
Finite Number Of Iterations

AbstractWe consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

Student attitudes towards linear algebra: an attempt to roll back the fog

2020 ◽  
Author(s):  
Barry J Griffiths ◽  
Samantha Shionis
Keyword(s):  
Student Perceptions ◽  
Linear Algebra ◽  
Student Attitudes ◽  
Linear Independence ◽  
Negative Emotions ◽  
Practical Applications ◽  
Key Concepts ◽  
The Subject ◽  
Roll Back

Abstract In this study, we look at student perceptions of a first course in linear algebra, focusing on two specific aspects. The first is the statement by Carlson that a fog rolls in once abstract notions such as subspaces, span and linear independence are introduced, while the second investigates statements made by several authors regarding the negative emotions that students can experience during the course. An attempt is made to mitigate this through mediation to include a significant number of applications, while continually dwelling on the key concepts of the subject throughout the semester. The results show that students agree with Carlson’s statement, with the concept of a subspace causing particular difficulty. However, the research does not reveal the negative emotions alluded to by other researchers. The students note the importance of grasping the key concepts and are strongly in favour of using practical applications to demonstrate the utility of the theory.

scholarly journals A converse to linear independence criteria, valid almost everywhere

2015 ◽  
Vol 38 (3) ◽  
pp. 513-528 ◽  
Author(s):  
S. Fischler ◽  
M. Hussain ◽  
S. Kristensen ◽  
J. Levesley
Keyword(s):  
Linear Independence ◽  
Almost Everywhere

scholarly journals Minimal Extensions of Algebraic Groups and Linear Independence

2001 ◽  
Vol 90 (2) ◽  
pp. 239-254 ◽  
Author(s):  
W.Dale Brownawell
Keyword(s):  
Linear Independence ◽  
Algebraic Groups

scholarly journals Linear independence of certain Lambert series

2014 ◽  
Vol 142 (10) ◽  
pp. 3411-3419 ◽  
Author(s):  
Florian Luca ◽  
Yohei Tachiya
Keyword(s):  
Linear Independence ◽  
Lambert Series

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

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