# Browse Classroom Capsules and Notes

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Displaying 81 - 85 of 85

The authors show that a function between vector spaces that maps lines to lines is either a collineation or has one-dimensional range.

The author shows that every $2 \times 2$ real matrix with nonreal eigenvalues represents the composition of the following three operations: (1) a vertical “lift” to a plane...

The author characterizes and counts the number of the many row-reduced echelon forms associated with the set of all $m \times n$ matrices.

An interesting formula for the determinant of the sum of any two matrices of the same size is presented. The formula can be used to obtain important results about the characteristic polynomial...

Given a square matrix $U$ and column vectors $\alpha$ and $\beta$, the author shows that $\det(U + \alpha \beta^T) = \det U + \beta^T$ Cof$(U) \alpha$.  This...