File Name: linear algebra geometry and transformation .zip
- Linear Algebra, Geometry and Transformation
- Understanding and Visualizing Linear Transformations
- Linear Algebra and Geometry
Linear Algebra, Geometry and Transformation
The aim of this chapter is to give an overview of the research that we have been conducting in our research group in Mexico about the linear transformation concept, focusing on difficulties associated with its learning, intuitive mental models that students may develop in relation with it, an outline of a genetic decomposition that describes a possible way in which this concept can be constructed, problems that students may experience with regard to registers of representation, and the role that dynamic geometry environments might play in interpreting its effects.
Preliminary results from an ongoing study about what it means to visualize the process of a linear transformation are reported. A literature review that directly relates to the content of this chapter as well as directions for future research and didactical suggestions are provided.
Linear transformation is one of the more abstract concepts studied in linear algebra. It is also one of the concepts with which students experience considerable difficulties Sierpinska ; Sierpinska et al.
Some of these difficulties may be related to their previously constructed function conceptions, since a linear transformation is a special kind of function between vector spaces. Trigueros and Bianchini observed that in the context of a modelling problem this relationship becomes clearer for students. Karrer and Jahn report other types of difficulties such as conversion between registers, especially from the graphic register to others; belief that linear transformations can only be applied to polygonal objects; and thinking that a transformation that conserves straight lines is necessarily linear.
These authors suggest the use of a dynamic geometry environment in which students can observe a linear transformation in three registers graphic, algebraic, and matrix simultaneously as well as the effect of making a change in one register on the others, in order to overcome these difficulties. In our research group, we have been studying the linear transformation concept from different angles, including how it is constructed, associated difficulties, conceptions that students might develop, and representations.
In this chapter the intention is to bring to the attention of an international audience selected work that has been conducted in Spanish about the learning of this notion. The findings reported here form part of a larger ongoing project about the understanding of Linear Algebra concepts. Although the data reported comes from studies conducted in Mexico and Chile, the observed phenomena might shed light on difficulties that students experience in other parts of the world as well.
The tendency to think in prototype transformations such as rotation, reflection, and dilation was present in all the students we interviewed. A similar phenomenon was reported in Sierpinska Some students were able to think in terms of compositions of these known transformations, but not beyond. However this is not to say that rotation is a simpler transformation than shear; it all depends on the context.
For example, Trigueros and Bianchini observed that students had a more difficult time finding the formula of a rotation since it contains trigonometric functions when working on a modelling problem than with the formula of a shear transformation; in this context rotation is considered more complex. As Hillel comments, not all linear transformations have simple geometric interpretations, even in two and three dimensional spaces.
However, for students these prototypes that are associated with certain movements in the plane replace the definition, giving rise to intuitive models that act as substitutes for mathematical theory Fischbein Another conception that we came across in this study is the one that associates a linear transformation to each vector instead of the whole plane.
A similar phenomenon was also observed in Dreyfus et al. Textbook illustrations that show one vector and its rotated image, in order to exemplify a rotation, for example, might contribute to students developing this viewpoint.
After this first study, we wanted to know whether these intuitive models prevailed only in geometric contexts or they were also present in algebraic contexts and, if so, in what way. Geometrically, we observed the same kinds of conceptions and intuitive models that students had developed as in the previous investigation.
Algebraically, though, this was not the case. In this context, algorithmic thinking dominated. Confrontation of different answers to the equivalent problems in geometric and algebraic contexts helped students to recognize their mistakes, but we are not sure to what extent it was helpful in constructing the concept.
Subsequently we wanted to understand how the linear transformation concept can be constructed in the mind of an individual. According to Arnon et al. According to our validated genetic decomposition, students with an Action conception can verify the linearity conditions for specific vectors and linear transformations but have difficulty in imagining the verification of the conditions for all the vectors of the domain and thinking about the concept of linear transformation in a general way.
When these actions get interiorized, they give rise to Processes related to the two conditions of linearity, which are then coordinated to construct a new Process that can be called the Process of linear transformation.
In this way, the sum and multiplication by scalar properties can be combined in a linear combination version, which unites them. This also shows the difference between the mathematical definition and the cognitive construction of a concept. Due to lack of previous research from an APOS perspective on the topic, in this study we took into consideration only the algebraic representation as a starting point.
This way students were able to realize how different components of a matrix contributed to the geometric effect of a linear transformation, especially in the case of stretch and shear.
Now, in order to illustrate the notion of mixing of registers, we give an example from our study in which during an interview the student Franco was working on the following problem adapted from Wawro Looking for a matrix that transforms the M on the left to the one on the right adapted from Wawro As a result of this study we note that in the literature there is a lack of consensus on the names and characteristics of registers of representation in linear algebra.
In fact, a variable vector and its variable image under the transformation can be placed on the screen simultaneously. In this situation, the effect of a transformation is directly observable, thus indirectly lending some visibility to the transformation itself. They also warn us against the pitfalls of computer environments and related designs, since students tend to develop conceptions of a linear transformation as the image of a vector as opposed to a function transforming the plane.
Interiorization toward Processes requires analyzing a sufficient quantity of repetitions in order to achieve an internal version of them; the sufficiency is reached when a significant reflection that permits taking control of the steps of the Actions is made.
Dynamic representations make a greater quantity of information available for students, practically in an immediate and continuous manner, through intuitive manipulations of representations. We propose that dynamic graphical representations work for the students as catalysts of the mechanism of interiorization.
In general, the Object conception is hard to reach and even after completing undergraduate courses, students do not show evidence of being able to apply actions on processes Arnon et al. Care was taken during instruction to provide students with activities that were intended to aid in the encapsulation of processes.
For example, after they had to come up with a linear transformation that sends a square region to itself, they had to modify it so that this time it would send the same square region to a bigger square or a general rectangle. Students in this study in general were able to develop an Object conception for linear transformations in the plane, but not outside of this context.
This points out the importance of providing students with the opportunity to work with transformations in different vector spaces. To the best of my knowledge, this was the first time dynamic geometry software was used in relation with research from the viewpoint of APOS theory.
Visualization of linear transformations has applications in computer graphics and robotics and in general where a study of geometrical representation of objects and their motion and transformation are involved. There have been different didactic suggestions in the literature as to how to visualize linear transformations. Monagan proposes both the use of CAS to display the images of different objects, such as circles, instead of the square regions that are normally used to illustrate the effect of a linear transformation and the use of animations to get an idea of the effect on the whole plane.
Triantafyllou and Timcenko recommend displaying the matrix of a linear transformation and a geometric object on the screen so that when changes are introduced in the matrix the effect on the object and on the whole grid covering the screen can be observed.
Hern and Long argue that three dimensions are necessary in order to study the effect of a linear transformation visually, making use of shapes such as a cube, a sphere, or a tetrahedron. All of these objects can be visualized geometrically—at least in the lower dimensions. It is quite a different matter to visualize the linear algebra processes which transform these objects, that is, linear transformations.
According to Piaget, visual perception which is the main tool used by Harel is possible for static objects, but not for dynamic processes. To visualize the latter, he argues, it is necessary to perceive a set of static phenomena and to reason about them in making mental constructions of dynamic processes. No matter how adequately we try to visualize the transformation of an arc into a straight line or vice versa, our images proceed in jumps and do no more Thus, images cannot exhaust the operation The results are: inability to anticipate in imagination, inadequate intermediate images and In the first approach, the emphasis is on the two static states, and in the latter approach the focus is on the dynamic movement, which starts with the initial state and ends in the final one.
Cognitively speaking, they imply the involvement of different structures or conceptions. There must be a clear sense that the beginning entity did not simply move to the new location ending entity , nor was it replaced by the new output ending entity , but that there was actually a metamorphosis of the beginning entity into the ending entity.
This has to do with imagining the process of linear transformation dynamically. He had to his disposition the GeoGebra software, which he made use of, as we will see. We wanted to find out how he was thinking about the linear transformation and if he made use of arguments about a continuum in his discussion of it. We will now go through his reasoning as he works on this problem. Image of the square in Fig. What would happen, instead of straight lines, We will leave the lines and we will use something more interesting.
We will sweep the plane with concentric circles. Shortly he realizes that the calculations would be quite messy and gives up on it. I: With what you have observed so far, can you describe how the transformation deforms the plane? I: OK, the image of the plane is a line under this transformation. But can you describe how it does it?
Luis: When you start approaching the identity line you start decreasing the norm, because the identity line is the kernel. I: Can you describe in which way the vectors of the plane are being transformed? I: We can determine the images of different objects. But can we describe how those images are being obtained? What is happening to the plane? Luis: I understand what you mean.
Luis: The only precise manner in which I showed how it happens is when I swept the plane with lines, point by point. Luis went back and forth between static-geometric he was drawing on the board , algebraic writing on paper , and dynamic-geometric software environments. He focused on objects and their images, through which the linear transformation started revealing itself. However he was reluctant to make further statements about how the deformation of the plane took place.
The visualization of the process may not be too difficult in the case of prototype transformations such as rotations and projections, but even with a slight increase in complexity, such as when two transformations are composed, it becomes considerably harder, as in this case where a projection and a dilation are involved. A variable vector has an unstable existence. Only while being dragged does it exist as such: a variable vector. When dragging stops, only a very partial record remains on the screen: An arrow with given, potentially variable length and direction.
The variability remains only potential, in the eye or mind of the beholder. If the student looking at the screen is not aware of this variability, the variable vector has ceased to exist as such. This is because the image rests on a spatialized imitation of operations which are themselves spatial. Piaget and Inhelder , p. Visualization is an act in which an individual establishes a strong connection between an internal construct and something to which access is gained through the senses.
Such a connection can be made in either of two directions. An act of visualization may consist of any mental construction of objects or processes that an individual associates with objects or events perceived by her or him as external.
Understanding and Visualizing Linear Transformations
Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Using geometric algebra for 3D linear transformations Abstract: Different methods exist for deriving a 3D linear transformation, including vector algebra, geometric algebra, and matrix algebra Artin, ; Hestenes, , Foley et al. In this paper, we introduce Ron Goldman's unified derivation method in vector algebra Goldman, combined with our developments in geometric algebra Hestenes, ; Dorst and Mann, ; Mann and Dorst, Our work includes using a unified method to derive various geometrical transformations, which are different from the current ray-tracing and transformation methods implemented in graphics algorithms and hardware Foley et al. We hope that these derivations extend the geometric algebra research and applications to the computer graphics field.
This book on linear algebra and geometry is based on a course given by renowned academician I. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics in affine and projective spaces , decomposition of finite abelian groups, and finitely generated periodic modules similar to Jordan normal forms of linear operators. Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics. Summing Up: Recommended.
Shafarevich, Alexey O. This is useful in developing the theory of iinear inequalities. This book is directed more at the former audience Readers should be able to apply the properties of determinants and matrix operations and solve linear systems of equations. These subjects include matrix algebra, vector spaces, eigenvalues and eigenvectors, symmetric matrices, linear transformations, and more. Chapter 4 defines the algebra of polynomials over a field, the ideals in that algebra, and the prime factorization of a polynomial. There are several details that distinguish this text from standard advanced linear algebra textbooks. It is used by the pure mathematician and by the mathematically trained scien-tists of all disciplines.
The Essentials of a First Linear Algebra Course and More Linear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear.
Linear Algebra and Geometry
Linear algebra is the branch of mathematics concerning linear equations such as:. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry , including for defining basic objects such as lines , planes and rotations. Also, functional analysis , a branch of mathematical analysis, may be viewed as basically the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering , because it allows modeling many natural phenomena, and computing efficiently with such models.
The aim of this chapter is to give an overview of the research that we have been conducting in our research group in Mexico about the linear transformation concept, focusing on difficulties associated with its learning, intuitive mental models that students may develop in relation with it, an outline of a genetic decomposition that describes a possible way in which this concept can be constructed, problems that students may experience with regard to registers of representation, and the role that dynamic geometry environments might play in interpreting its effects. Preliminary results from an ongoing study about what it means to visualize the process of a linear transformation are reported. A literature review that directly relates to the content of this chapter as well as directions for future research and didactical suggestions are provided. Linear transformation is one of the more abstract concepts studied in linear algebra. It is also one of the concepts with which students experience considerable difficulties Sierpinska ; Sierpinska et al.
In Section 2. Linear algebra is one of the key mathematical pillars underlying much of the work that we do in deep learning and in machine learning more broadly. While Section 2.
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated. The objective is to collect and collate metadata and provide full text index from several national and international digital libraries, as well as other relevant sources. It is a academic digital repository containing textbooks, articles, audio books, lectures, simulations, fiction and all other kinds of learning media. Show full item record. Linear Algebra Geometry and Transformation. Solomon, Bruce.
Home Forum Login. Download PDF. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the spectral theorem. The text starts with basic questions about images and pre-images of mappings, in- jectivity, surjectivity, and distortion. In the process of answering these questions in the linear seting, the bok covers al the standard topics for a frst course on linear algebra, Linear Algebra including linear systems, vector geometry, matrix algebra, subspaces, independence, dimension, orthogonality, eigenvectors, and diagonalization.
We have moved out of the familiar world of functions of one variable: we are now thinking about functions that transform a vector into a vector. Then we have.