Linear Algebra
https://maa.org/taxonomy/term/42292/all
enCollapsed Matrices with (almost) the Same Eigenstuff
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/collapsed-matrices-with-almost-the-same-eigenstuff
<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 author describes a method for constructing a smaller matrix with the same (or similar) eigenvalues that would be usable in the classroom. He illustrates this with matrices for Leslie population models.</em></p>
</div></div></div>The Arithmetic of Algebraic Numbers: An Elementary Approach
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-arithmetic-of-algebraic-numbers-an-elementary-approach
<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>If \(r\) and \(s\) are algebraic numbers, then \(r + s\), \(rs\), and \(r/s\) are also algebraic. The proof provided in this capsule uses the ideas of characteristic polynomials, eigenvalues, and eigenvectors.</em></p>
</div></div></div>Rank According to Perron: A New Insight
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/rank-according-to-perron-a-new-insight
<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>Suppose we have several alternatives that we wish to rank. For example, we may wish to rank five teachers according to their teaching excellence. The author constructs a positive matrix \(A\) based on pairwise comparisons of the alternatives, and uses the Perron principal eigenvector to find a ranking. The author employs dominance walks to obtain these results.</em></p>
</div></div></div>On the Measure of Solid Angles
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/on-the-measure-of-solid-angles
<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 author revisits formulas of measuring solid angles that he could find only in centuries-old literature, and provides modern versions of the proofs.</em></p>
</div></div></div>Approaches to the Formula for the \(n\)th Fibonacci Number
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/approaches-to-the-formula-for-the-nth-fibonacci-number
<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>In this capsule, proofs of the equivalence of the two definitions of the Fibonacci numbers are discussed. This helps the undergraduate view mathematics as a unified whole with a variety of techniques.</em></p>
</div></div></div>Image Reconstruction in Linear Algebra
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/image-reconstruction-in-linear-algebra
<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 main object is to solve the inverse problem of recovering the original scene, represented by a vector or a matrix, from its photograph, represented by a product of a matrix and the original vector or matrix. The solution of the resulting matrix equation gives rise to the reconstruction of the original scene</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>A Surprise from Geometry
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-surprise-from-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>Consider \(n\) vectors issuing from the origin in \(n\)-dimensional space. The author shows that the statement “any set of \(n\) vectors in \(n\)-space, no two of which meet at greater than right angles, can be rotated into the non-negative orthant” is true for \(n \leq 4\), but false for \(n>4\).</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>Parametric Equations and Planar Curves
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/parametric-equations-and-planar-curves
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The author shows that the system characterized by</p>
<p>\(x(t) = a_1 t^2 + b_1 t + c_1\)<br />
\(y(t) = a_2 t^2 + b_2 t + c_2 \)<br />
\(z(t) = a_3 t^2 + b_3 t + c_3\)</p>
<p>must lie in a plane. He does this with an illustrative example represented by<br />
\(x(t) = A [t^2, t, 1]^T\)<br />
where \(A\) is a constant \(3 \times 3\) matrix.<br />
</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>Two by Two Matrices with Both Eigenvalues in \(Z/pZ\)
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/two-by-two-matrices-with-both-eigenvalues-in-zpz
<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>This article provides a non-group theory approach to finding the number of two by two matrices over \( Z/pZ\) that have both eigenvalues in the same field. The strategy is to use the quadratic formula to find the roots of the characteristic polynomial of a matrix and then count the number of matrices for which these roots are in \(Z/pZ \).</em></p>
</div></div></div>A Transfer Device for Matrix Theorems
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-transfer-device-for-matrix-theorems
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>The author presents a method to transfer matrix identities over the real numbers to matrix identities over an arbitrary commutative ring. Several examples are given, including \(\det(AB)= \det(A) \det(B) \), the Cayley-Hamilton Theorem, and identities involving adjoint matrices.</p>
</div></div></div>Tennis with Markov
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/tennis-with-markov
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>In the game of tennis, if the probability that player \(A\) wins a point against player \(B\) is a constant value \(p\), then the probability that \(A\) will win a game from deuce is \(p^2/(1 - 2p + 2p^2)\). This result has been obtained in a variety of ways, and the authors use a formal Markov chain approach to derive it.</p>
</div></div></div>Matrix Patterns and Undetermined Coefficients
https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrix-patterns-and-undetermined-coefficients
<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 Method of Undetermined Coefficients is approached by using linear operators and making use of patterns associated with matrix multiplication. The authors discuss several pedagogical advantages to this approach.</em></p>
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