# Browse Classroom Capsules and Notes

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Displaying 21 - 30 of 85

The author revisits formulas of measuring solid angles that he could find only in centuries-old literature, and provides modern versions of the proofs.

The author classifies the quadratic forms defined by simple 2 by 2 matrices and illustrates them with corresponding quadratic surfaces.

A closed form of the Wronskian for $sin(kx)$ and for $e^{kx}, k=1,2,\ldots,n$ is obtained. The derivation depends on trigonometric identities and properties of the determinant....

An $n \times n$ matrix whose rows, columns, and diagonal all sum to the same number $m$ is called magic, and the number $m$ is called the magic sum.  If $A$ is a magic square matrix...

The author gives an expression for $\pi$ involving an infinite sequence of determinants, each representing the area of a triangle.

Using mathematical induction, the author provides a simple proof that the reduced row echelon form of a matrix is unique.

Consider the vector space of polynomials of degree less than $n$, and a polynomial $p(x)$ in this space. The author describes the matrix $M(r)$ that maps the polynomial $p(x)$ to \(p(...

A short elementary proof of the equality of row rank and column rank is given. The proof requires only the definition of matrix multiplication and the fact that a minimal spanning set is a...

Newton’s identities relate the coefficients of a polynomial to sums of powers of its roots.  The author uses the Cayley-Hamilton theorem and properties of the trace of a matrix to...

Two least-squares methods are used to estimate city and highway gas mileage from readily measured data. First, the author uses the standard least squares method which is suitable for a first...