scholarly journals Appendix F: Review of Linear Algebra: Orthogonality, Basis, Linear Independence, and the Gram–Schmidt Algorithm

2004 ◽  
pp. 537-541
Keyword(s):  
Linear Algebra ◽  
Linear Independence

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.

Geometric Mappings on Geometric Lattices

1971 ◽  
Vol 23 (1) ◽  
pp. 22-35 ◽  
Author(s):  
David Sachs
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Classical Result ◽  
Linear Independence ◽  
Closure Operator ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Algebraic Methods ◽  
Linear Transformations ◽  
Geometric Lattices

It is a classical result of mathematics that there is an intimate connection between linear algebra and projective or affine geometry. Thus, many algebraic results can be given a geometric interpretation, and geometric theorems can quite often be proved more easily by algebraic methods. In this paper we apply topological ideas to geometric lattices, structures which provide the framework for the study of abstract linear independence, and obtain affine geometry from the mappings that preserve the closure operator that is associated with these lattices. These mappings are closely connected with semi-linear transformations on a vector space, and thus linear algebra and affine geometry are derived from the study of a certain closure operator and mappings which preserve it, even if the “space” is finite.

Notation and Basic Operations

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Stochastic Processes ◽  
Linear Algebra ◽  
Linear Independence ◽  
Unit Vector ◽  
Inner Product ◽  
Matrix Product ◽  
Outer Product ◽  
Vector Addition ◽  
Vector Matrix

This chapter introduces some of the basic players of the algebraic drama about to unfold, and the uniform notations used in this book. While Chapter 3 is a more formal introduction to linear algebra, this introductory chapter presents some of the most basic elements, and permitted manipulations and operations, of linear algebra. It covers the following scalar variables, stochastic processes and variables, matrix variables, fields, vector variables, vector transpose, vector addition, linear independence, inner product of two vectors, projection, orthogonality, the norm of a vector, unit vector, matrix variables, matrix addition, transpose of a matrix, special matrices, matrix product, and matrix outer product.

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