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, ...

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 ...

Let $v_1, \dots, v_m$ denote a basis of $V$. The dimension of $\operatorname{range} T$ is the same as the dimension of $\operatorname{span}(Tv_1, \dots, Tv_m)$. So if ...

which holds purely because composing with an identity map doesn’t change anything. Now apply Theorem 4.19.1 from the previous section twice: you get the change of basis formula: In this subsection ...

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 ...

To provide students with an introduction to modern abstract linear algebra. Building on their existing knowledge of matrix methods, students will experience the benefits of the abstract and rigorous ...

Algebraic structures and linear maps form a cornerstone in modern mathematics, underpinning areas as diverse as abstract algebra and functional analysis. Algebraic structures such as groups, rings, ...

The aim of this paper is to show that there are no piecewise linear maps of type 2∞. For this purpose we use the fact that any piecewise monotone map of type 2∞ has an infinite ω-limit set which is a ...

Abstract: The linear one-turn map of a storage ring contains coupling information on which a correction algorithm can be based. In principal, the one-turn matrix can be fitted from turn-by-turn data ...