Linear Algebra
https://www.maa.org/taxonomy/term/42292/all
enMath Bite: On the Definition of Collineation
https://www.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>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94761 at https://www.maa.orgObtaining the \(QR\) Decomposition by Pairs of Row and Column Operations
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/obtaining-the-qr-decomposition-by-pairs-of-row-and-column-operations
<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 gives an alternative to the usual Gram-Schmidt process and shows how to obtain the “QR Factorization” by using pairs of row column operations.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95053 at https://www.maa.orgComputing the Fundamental Matrix for a Reducible Markov Chain
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/computing-the-fundamental-matrix-for-a-reducible-markov-chain
<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>A Markov chain with 9 states is used to illustrate a technique for finding the fundamental matrix.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95376 at https://www.maa.orgDoes the Generalized Inverse of \(A\) Commute with \(A\)?
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/does-the-generalized-inverse-of-a-commute-with-a
<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>Not all square matrices commute with their generalized inverse (Moore-Penrose inverse). The author gathers equivalent conditions for the generalized inverse of a matrix to commute with the matrix itself. Then he shows that, in this case, the generalized inverse may be represented as a polynomial in the given matrix.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95504 at https://www.maa.orgA Nonstandard Approach to Cramer's Rule
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-nonstandard-approach-to-cramers-rule
<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>Cramer's Rule gives an explicit formulation for the unique solution to a system of \(n\) equations in \(n\) unknowns when the coefficient matrix of the system is invertible. The standard proof is developed using the adjoint matrix. In this capsule, the author uses properties of determinants and general matrix algebra to provide an alternative proof of Cramer's Rule.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95773 at https://www.maa.orgDeterminantal Loci
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/determinantal-loci
<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 characterizes the points \((x, y)\) in the plane for which the determinant of a matrix of a particular form involving \((x, y)\) is \(0\). The matrices of interest have the form \(A+xL+uM\), where \(A\), \(L\), and \(M\) are square matrices, \(L\) and \(M\) are of rank one, and \(L + M\) is of rank two.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94792 at https://www.maa.orgA Geometrical Approach to Cramer's Rule
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-geometrical-approach-to-cramers-rule
<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>Motivated by the interpretation of a determinant of a \(2 \times 2\) matrix as the area of a parallelogram, the author derives Cramer's rule geometrically.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95056 at https://www.maa.orgProof without Words: A 2 x 2 Determinant Is the Area of a Parallelogram
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/proof-without-words-a-2-x-2-determinant-is-the-area-of-a-parallelogram
<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 presents a visual proof that the determinant of a 2 by 2 matrix equals the area of the corresponding parallelogram.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95382 at https://www.maa.orgMatrices, Continued Fractions, and Some Early History of Iteration Theory
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/matrices-continued-fractions-and-some-early-history-of-iteration-theory
<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>Continued fractions of the form \( \frac{1}{1 + \frac{c}{1 + \frac{c}{ 1 +\ddots}}} \) are analyzed using linear algebra and iteration theory. The continued fractions of interest are closely related to a class of \(2 \times 2\) matrices, and the eigenvalues and eigenvectors of those matrices are investigated to determine when the corresponding continued fractions converge. Historical references are included.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95514 at https://www.maa.orgAn Alternate Proof of Cramer's Rule
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/an-alternate-proof-of-cramers-rule
<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 provides a short proof of Cramer’s rule that avoids using the adjoint of a matrix.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95777 at https://www.maa.orgGaussian Elimination in Integer Arithmetic: An Application of the \(L\)-\(U\) Factorization
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/gaussian-elimination-in-integer-arithmetic-an-application-of-the-l-u-factorization
<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>Using L-U factorization, the author generates examples of matrices for which Gaussian elimination process can be done in integer arithmetic, including examples of matrices that are invertible over the integers.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com94804 at https://www.maa.orgA Diagonal Perspective on Matrices
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/a-diagonal-perspective-on-matrices
<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 present real matrices from a diagonal perspective, to supplement the usual row/column perspective and to offer contexts in which this is a useful mode.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95090 at https://www.maa.orgUsing Quadratic Forms to Correct Orientation Errors in Tracking
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/using-quadratic-forms-to-correct-orientation-errors-in-tracking
<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 noise in data transmission produces a not quite orthogonal matrix that is known to be orthogonal, how does one find the "nearest" orthogonal matrix? This capsule recasts the problem as one of maximizing a quadratic form on the four-dimensional unit sphere, and sketches a solution.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95396 at https://www.maa.orgHow to Determine Your Gas Mileage
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/how-to-determine-your-gas-mileage
<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>Two least-squares methods are used to estimate city and highway gas mileage from readily measured data. First, the author uses the standard least squares method which is suitable for a first linear algebra course. Second, the author discusses a more accurate weighted least squares method.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95518 at https://www.maa.orgApropos Predetermined Determinants
https://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/apropos-predetermined-determinants
<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 illustrates how certain determinants can be used to motivate students. The determinants in question have terms in artithmetic progressions.</em></p>
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Fri, 12 Jul 2013 15:03:25 +0000
hosted@newtarget.com95783 at https://www.maa.org