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Browse Classroom Capsules and Notes

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Displaying 11 - 20 of 85

Approaches to the Formula for the \(n\)th Fibonacci Number

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...

Parametric Equations and Planar Curves

The author shows that the system characterized by

\(x(t) = a_1 t^2 + b_1 t + c_1\)
\(y(t) = a_2 t^2 + b_2 t + c_2 \)
\(z(t) = a_3 t^2 + b_3 t + c_3\)

must lie in a...

Matrix Patterns and Undetermined Coefficients

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...

The Gunfight at the OK Corral

This article uses a generalization of a three-way gunfight to motivate the construction and solution to a first order linear system of difference equations.  The method of undetermined...

Math Bite: On the Definition of Collineation

The authors show that a function between vector spaces that maps lines to lines is either a collineation or has one-dimensional range.

Determinantal Loci

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...

Gaussian Elimination in Integer Arithmetic: An Application of the \(L\)-\(U\) Factorization

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...

A Geometric Intrepretation of the Columns of the (Pseudo) Inverse of A

The pseudo inverse of a matrix \(A\) is used to obtain information about the rows of \(A\).

Classifying Row-Reduced Echelon Matrices

The author characterizes and counts the number of the many row-reduced echelon forms associated with the set of all \(m \times n\) matrices.

Determinants of Sums

An interesting formula for the determinant of the sum of any two matrices of the same size is presented. The formula can be used to obtain important results about the characteristic polynomial...

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