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 ...

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, ...

ABSTRACT: We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form: . We study some special type of factorization for plus-operators T, among ...

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 ...