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Matrices
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
Linear algebra is the branch of mathematics that deals with vector spaces and linear transformations. For someone just beginning their study of linear algebra, that is probably a meaningless statement. It does, however, convey the idea that there are two concepts in our study that will be of utmost importance; namely, vector spaces and linear transformations. A primary tool in our study of these topics will be matrices. In this chapter we give the rules that govern matrix algebra.
Vectors and linear systems
Published in Qingwen Hu, Concise Introduction to Linear Algebra, 2017
A central goal of linear algebra is to solve systems of linear equations. We have seen the simplest linear equation ax = b, where x∈R $ x \in {\mathbb{R}} $ (the symbol “∊ ” means “in”) is the unknown variable and a, b∈R $ b \in {\mathbb{R}} $ are constants. It is known that there are three scenarios for the solutions: 1) if a ≠ 0, there is a unique solution x=ba;2 $ x = \frac{b}{a};2 $ 2) if a = 0, b ≠ 0, there is no solution; 3) if a = b = 0, there are infinitely many solutions. We are then motivated to investigate systems of equations with multiple unknown variables. The following system
Vectors, Matrices, and Linear Systems
Published in Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong, Intelligent Diagnosis and Prognosis of Industrial Networked Systems, 2017
Chee Khiang Pang, Frank L. Lewis, Tong Heng Lee, Zhao Yang Dong
In linear algebra, linear transformations can be represented by matrices. If T is a linear transformation mapping ℝn → ℝm where x ∈ ℝn and y ∈ ℝm are vectors, then T :→ y = Ax for some m × n matrix A. Here, A is called the transformation matrix.
Enhancing pre-service mathematics teachers’ understanding of function ideas
Published in International Journal of Mathematical Education in Science and Technology, 2022
Hande Gülbağci Dede, Zuhal Yilmaz, Hatice Akkoç, David Tall
Participants were a convenience sample of PMTs studying in a teacher preparation programme at a state university in Istanbul, Turkey. The first cycle of the module was completed with 17 PMTs who enrolled in the Mathematics Teaching I and II courses. The PMTs’ learning experiences concerning the concept of function in high school were as follows: (a) PMTs were introduced to functions as a special kind of a relation at 9th grade, (b) the national mathematics curriculum and mathematics textbooks at that time privileged ideas of mapping, input-output, algebra, and rule, (c) there was no learning objective in this curriculum that focused on relations between variables or covariation. Before the module, the PMTs took sixteen mathematics courses such as calculus, linear algebra, and analytic geometry. Also, all participants took mathematics education courses including geometry teaching, algebra teaching, and mathematics curriculum.
Tensor calculus: unlearning vector calculus
Published in International Journal of Mathematical Education in Science and Technology, 2018
Wha-Suck Lee, Johann Engelbrecht, Rita Moller
In Section 3, we introduce the idea that a coordinate transformation is a dualism in the sense of seeing the same object through the different lenses of a different coordinate systems. We present the notion of the Jacobian as a coordinate transformation that has been linearized by differentiation. The Jacobian is a local linear substitute to the original coordinate transformation which needs not be linear. This coordinate transformation approach then adds a geometric flavour to the well-drilled concept of the determinant of a Jacobian as a fudge factor which is the limit of the ratio of the area of the image set (under the coordinate transformation) to the area of the pre-image set. Furthermore, the Jacobian has the critical property of preserving derivatives from one coordinate system to another. Thus, a Jacobian is a linear dualism map between the two dual realities of coordinate systems with the peculiar property of preserving tangents. This property of preserving of tangents explains the fundamental role of the Jacobian in tensor calculus. In tensor calculus, objects are treated as derivatives. In Section 4, we show that even the fundamental objects of linear algebra, coordinate basis vectors, are derivatives. The change-of-basis matrix in linear algebra is also a derivative (Proposition 3.1). So, in this sense, calculus (differentiation) is more fundamental than linear algebra.
Fifty years of similarity relations: a survey of foundations and applications
Published in International Journal of General Systems, 2022
Positive definite and semi-definite matrices appear in many branches of Mathematics such as in the study of quadratic forms, optimization problems and classification of quadric (hyper)-surfaces; but what is more important in Artificial Intelligence is that in linear algebra the matrix associated to an inner product of a vector space is positive definite. In this case, a Euclidean distance can be defined on this space. In many AI problems, in particular in machine learning, vision or cluster analysis, a distance on a universe X is needed. In these cases, the use of kernels on X allows us to define a Euclidean-like distance d on X with all its good properties (Pkalska and Duin 2005; Moser 2006b).