# Classroom Capsules and Notes

Capsules By Courses. We are organizing the capsules into courses, when possible using the same topics as are used in Course Communities. So far we have organized capsules for the following courses:

You may select topics within each course.

Notes: Sequences and Series is part of One-Variable Calculus. We felt that since this topic had so many capsules associated with it, we wanted to introduce sub-topics. Also, the Number Theory collection of capsules does not correspond to a course in Course Communities, but has topics selected by the Editorial Board for Classroom Capsules and Notes.

## Featured Items

##### An Intuitive Proof of the Singular Value Decomposition of a Matrix

Using a simple trigonometric limit, the author provides an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

##### A Tricky Linear Algebra Example

In this article a classroom "trick" involving square arrangements of natural numbers is used to motivate a discussion of a special class of matrices. In particular, a basis is obtained for those $$n$$ by $$n$$ square matrices with the property that if $$n$$ entries are selected from the matrix so that no two values are in the same row or the same column, then the sum of these $$n$$ entries will always be the same.

##### Finding Matrices that Satisfy Functional Equations

The author shows how to solve a class of analytic functions using an approach demonstrating a surprising connection between multivariable calculus and linear algebra.

##### Trick or Technique?

The author studies the functions whose integral can be evaluated by an algebra trick using one-step or two-step integration by parts.

##### 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 $$\mathbb{Z}/p\mathbb{Z}$$ 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 $$\mathbb{Z}/p\mathbb{Z}$$.

##### On Laplace's Extension of the Buffon Needle Problem

The classical Buffon needle problem is to find the probability that a needle of length $$n$$ when dropped on a floor made of boards of width $$b$$ will cross a crack between the boards.