(ST)(T^{-1}S^{-1}) = STT^{-1}S^{-1} = SS^{-1} = I Hence, $ST$ is invertible and $(ST)^{-1} = T^{-1}S^{-1}$. _Exercise 2_ Let $N$ be the set of noninvertible operators ...

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, ...

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 ...

Abstract: This book contains a detailed discussion of the matrix operation, its properties, and its applications in finding the solution of linear equations and determinants. Linear algebra is a ...

Let 𝓤 be a unital ★-algebra and δ : 𝓤 → 𝓤 be a linear map behaving like a derivation or an anti-derivation at the following orthogonality conditions on elements of 𝓤: xy = 0, xy★ = 0, xy = yx = 0 ...

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 ...

This repository contains manual solutions to exercises from Sheldon Axler's Linear Algebra Done Right (4th Edition). Please be aware that there may be errors or typos in the solutions. Contributions ...

This paper proposes the novel algebraic structure of a linear ring space. A linear ring space is an order triad consisting of two rings, and a linear map between the two rings. The definition of quasi ...