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Browse Classroom Capsules and Notes

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Displaying 71 - 80 of 85

Clock Hands Pictures for \(2 \times 2\) Real Matrices

The article illustrates the notion of eigenvalue and its corresponding eigenvector using hands of an analog clock. This capsule deals with \(2 \times 2\) real matrices, single eigenvalues, and...

Notational Collisions

This capsule points out several potential confusions in commonly used linear algebra notation.

The Matrix of a Rotation

Given a unit vector \(p\) in \( \mathbf{R}^3\) and an angle \( \theta\), what is the matrix of the rotation of \(\mathbf{R}^3\) about \(p\) through an angle of \(\theta\) in terms of 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 \(A\).

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...

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...

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...

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