Linear Transformation
https://maa.org/taxonomy/term/42298/all
enThe Matrix of a Rotation
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-matrix-of-a-rotation
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Given a unit vector \(p\) in \( \mathbf{R}^3\) and an angle \( \theta\), what is the matrix of the rotation of \(\mathbf{R}^3\) about \(p\) through an angle of \(\theta\) in terms of the standard basis? The author obtains an explicit matrix without changing bases.</em></p>
</div></div></div>A Geometric Approach to Linear Functions
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-geometric-approach-to-linear-functions
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>There are three somewhat distinct topics in this article: the condition for linear functions to commute, a linear function as a transformation of the number line, and linear difference equations. A linear function \(y=f(x)=ax+b\) can be characterized in terms of slope and the “center of reflection,” both of which reflect the geometric property of the function. </em></p>
</div></div></div>The Axis of a Rotation: Analysis, Algebra, Geometry
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-axis-of-a-rotation-analysis-algebra-geometry
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Given any 3 by 3 rotation matrix \(A\) (i.e. orthogonal with determinant \(1\) and an arbitrary vector \(x\), the vector \( Ax +A^{T}x+[1−\)tr\((A)]x\) lies on the axis of rotation. The article provides three different approaches, requiring various levels of background knowledge, to prove and/or explain the given result.</em></p>
</div></div></div>Polynomial Translation Groups
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/polynomial-translation-groups
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Consider the vector space of polynomials of degree less than \(n\), and a polynomial \(p(x)\) in this space. The author describes the matrix \(M(r) \) that maps the polynomial \(p(x)\) to \(p(x+r)\), where \(r\) is a real number. The group structure of the matrices \(M(r)\) under multiplication then gives rise to various combinatorial identities.</em></p>
</div></div></div>Root Preserving Transformations of Polynomials
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/root-preserving-transformations-of-polynomials
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The article answers negatively the question, “Is there a (non-trivial) linear transformation \(T\) from \(P_n\), the vector space of all polynomials of degree at most \(n\), to \(P_n\) such that for each \(p\) in \( P_n\) with a real or complex root, the polynomials \(p\) and \(T( p)\) have a common root?</em>" <em>The proof is based on the fact polynomials of degree at most \(n\) have at most \(n\) roots in the real or complex numbers. This article investigates an area common to algebra and linear algebra.</em></p>
</div></div></div>Math Bite: On the Definition of Collineation
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/math-bite-on-the-definition-of-collineation
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>The authors show that a function between vector spaces that maps lines to lines is either a collineation or has one-dimensional range.</em></p>
</div></div></div>Additivity + Homogeneity
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/additivity-homogeneity
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Linear transformations satisfy properties of both additivity and homogeneity. This capsule presents classes of functions that satisfy additivity but not homogeneity and vice versa.</em></p>
</div></div></div>Linear Transformation of the Unit Circle in \(\Re^2\)
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/linear-transformation-of-the-unit-circle-in-re2
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><em>Instead of using the image of a unit square in studying linear transformations in \(R^2\), the authors show that looking at images of the unit circle yield an informative picture and illustrate several basic ideas.</em></p>
</div></div></div>Linear Transformations in the Plane
https://maa.org/programs/faculty-and-departments/course-communities/linear-transformations-in-the-plane
Conceptual Linear Algebra Online
https://maa.org/programs/faculty-and-departments/course-communities/conceptual-linear-algebra-online
Linear transformation applet
https://maa.org/programs/faculty-and-departments/course-communities/linear-transformation-applet-0
Online Math Lab at UC Santa Barbara
https://maa.org/programs/faculty-and-departments/course-communities/online-math-lab-at-uc-santa-barbara
Computer Graphics Project
https://maa.org/programs/faculty-and-departments/course-communities/computer-graphics-project
Matrix Algebra Demos
https://maa.org/programs/faculty-and-departments/course-communities/matrix-algebra-demos
Linear Algebra Toolkit
https://maa.org/programs/faculty-and-departments/course-communities/linear-algebra-toolkit-0