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Browse Classroom Capsules and Notes

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Displaying 61 - 70 of 85

Linear Algebra in the Financial World

Linear algebra is used to study financial trading strategies and expectations. Financial conditions are examined via matrix equations, using rank, column space, and null space arguments....

Eigenvalues of Matrices of Low Rank

A technique is discussed for finding the eigenvalues of square matrices of small rank, which is useful for student discovery in a linear algebra class. The eigenvalues of a matrix of rank 1 or...

Collapsed Matrices with (almost) the Same Eigenstuff

The author describes a method for constructing a smaller matrix with the same (or similar) eigenvalues that would be usable in the classroom. He illustrates this with matrices for Leslie...

The Arithmetic of Algebraic Numbers: An Elementary Approach

If \(r\) and \(s\) are algebraic numbers, then \(r + s\), \(rs\), and \(r/s\) are also algebraic. The proof provided in this capsule uses the ideas of characteristic polynomials, eigenvalues,...

The Existence of Multiplicative Inverses

Using only basic ideas from linear algebra and number theory, the authors show that if \(c\) is square-free, the ring \(Q [\sqrt[n]{c}] \) is a field. An arbitrary nonzero element of the...

A Diagonal Perspective on Matrices

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.

Generating Exotic-Looking Vector Spaces

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

A Picture is Worth a Thousand Words

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...

Finding a Determinant and Inverse Matrix by Bordering

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...

When is Rank Additive?

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...

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