# Browse Classroom Capsules and Notes

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Displaying 41 - 50 of 85

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...

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,...

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...

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 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 $2 \times 2$...

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...

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...

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

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

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...