I need to prove that the transvection matrices generate the special linear group $\operatorname{SL}_n \left(\mathbb{R}\right) $. I want to proceed using induction on n. I was able to prove the 2×2 case, but I am having difficulty with the n+1 case. I supposed that the elementary matrices of the first type generate $SL_n(\mathbb{R})$. And I […]

If $a \in S$ is some invertible element in a ring $S$, then a computation shows $$\pmatrix{a & 0 \\ 0 & a^{-1}} = \pmatrix{1 & a \\ 0 & 1} \pmatrix{1 & 0 \\ -a^{-1} & 1} \pmatrix{1 & a \\ 0 & 1} \pmatrix{0 & -1 \\ 1 & 0}.$$ If $R \to […]

I’m reading some stuff about algebraic K-theory, which can be regarded as a “generalization” of linear algebra, because we want to use the same tools like in linear algebra in module theory. There are a lot of open problems and conjectures in K-theory, which are “sometimes” inspired by linear algebra. So I just want to […]

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