During my second course in linear algebra, I encountered a frequent need to convert linear maps into matrices. This task became particularly challenging when working with complex maps and bases, ...

1 Department of Mathematical Sciences, Mathematical Finance and Econometrics, Catholic University of the Sacred Heart, Milan, Italy 2 Department of Economics, University of Bamberg, Bamberg, Germany ...

Consider $T = (x_1,..,x_5) \to (x_4, x_5)$. We have $\operatorname{null} T = \{(x_1,x_2,x_3,0,0) | x_1,x_2,x_3 \in \mathbb{F}\}$, and $\operatorname{range} T ...

Linear maps are abstractly defined things. We’d like to make them concrete. We do this by making the following observation: once you know what a linear transformation does on a basis, you know what it ...

1 Department of Computer Science, Colorado State University, Fort Collins, CO, United States 2 Department of Mathematics, Colorado State University, Fort Collins, CO, United States In the ...

But an arbitrary function between two vector spaces doesn’t necessarily give you any information about their relationship as vector spaces. To get such information, we need to restrict to functions ...

Transfer matrix methods are widely used to calculate properties of particle orbits as they pass through linear beam elements such as drif t spaces, bending magnets, and quadrupoles. A new method of ...