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Browse Classroom Capsules and Notes

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Displaying 51 - 60 of 85

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.

Obtaining the \(QR\) Decomposition by Pairs of Row and Column Operations

The author gives an alternative to the usual Gram-Schmidt process and shows how to obtain the “QR Factorization” by using pairs of row column operations.

Elementary Linear Algebra and the Division Algorithm

This article provides a proof of division algorithm in polynomial rings using linear algebra techniques. The proof uses the fact that polynomials of degree equal to and less than n form a...

A Geometrical Approach to Cramer's Rule

Motivated by the interpretation of a determinant of a \(2 \times 2\) matrix as the area of a parallelogram, the author derives Cramer's rule geometrically.

Using Consistency Conditions to Solve Linear Systems

This article provides a method for solving general linear systems via row reduction on special augmented matrices. This approach can avoid back substitution, but requires bigger matrices....

Additivity + Homogeneity

Linear transformations satisfy properties of both additivity and homogeneity.  This capsule presents classes of functions that satisfy additivity but not homogeneity and vice versa....

Image Reconstruction in Linear Algebra

The main object is to solve the inverse problem of recovering the original scene, represented by a vector or a matrix, from its photograph, represented by a product of a matrix and the...

Linear Transformation of the Unit Circle in \(\Re^2\)

Instead of using the image of a unit square in studying linear transformations in \(R^2\), the authors show that looking at images of the unit circle yield an informative picture and...

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