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Browse Classroom Capsules and Notes

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Displaying 31 - 40 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.

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....

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

The paper gives a short proof that for any \(n\) x \(n\) matrices \(A\) and \(C\) over a field of scalars, \(AC = I\) if and only if \(CA = I\). The proof relies on familiarity with...

The article answers negatively the question, “Is there a (non-trivial) linear transformation \(T\) from \(P_n\), the vector space of all polynomials of degree at most \(n\), to \(P_n\)...

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

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

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 Cayley-Hamilton theorem may be used to determine explicit formulae for all the square roots of \(2 \times 2\) matrices. These formulae indicate exactly when a \(2 \times 2\)...

Given any 3 by 3 rotation matrix \(A\) (i.e. orthogonal with determinant \(1\) and an arbitrary vector \(x\), the vector \( Ax +A^{T}x+[1−\)tr\((A)]x\) lies on the axis of rotation. The...

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