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Browse Classroom Capsules and Notes

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Displaying 31 - 40 of 85

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.

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.

The authors present real matrices from a diagonal perspective, to supplement the usual row/column perspective and to offer contexts in which this is a useful mode.

The authors present a procedure for finding the determinant and inverse of a special class of matrices.  The strategy adds borders to the original matrix, and makes use of row operations...

This capsule presents necessary and sufficient conditions for the matrix rank of a sum to be the sum of the ranks.  The crux of the argument uses the fact that the rank of a matrix is the...

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...

This note describes how to generate exercises allowing students to study nonstandard operations on familiar objects.

The author provides geometric illustrations of four subspaces associated with a matrix. Thinking of a matrix as a map between real vector spaces, the illustrations motivate the decomposition...

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...

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

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