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Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 85

The Existence of Multiplicative Inverses

Using only basic ideas from linear algebra and number theory, the authors show that if \(c\) is square-free, the ring \(Q [\sqrt[n]{c}] \) is a field. An arbitrary nonzero element of the...

The Arithmetic of Algebraic Numbers: An Elementary Approach

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,...

A Geometric Approach to Linear Functions

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...

Two by Two Matrices with Both Eigenvalues in \(Z/pZ\)

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...

The Square Roots of \(2 \times 2\) Matrices

The Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of \(2 \times 2\) matrices. These formulae indicate exactly when a \(2 \times 2\)...

The Axis of a Rotation: Analysis, Algebra, Geometry

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...

A Note on the Equality of the Column and Row Rank of a Matrix

An elementary argument, different from the usual one, is given for the familiar equality of row and column rank. The author takes “full advantage of the following two elementary...

Computing the Fundamental Matrix for a Reducible Markov Chain

A Markov chain with 9 states is used to illustrate a technique for finding the fundamental matrix.

Proof without Words: A 2 x 2 Determinant Is the Area of a Parallelogram

The author presents a visual proof that the determinant of a 2 by 2 matrix equals the area of the corresponding parallelogram.

Using Quadratic Forms to Correct Orientation Errors in Tracking

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...

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