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Let $\textsf V$ be a finite-dimensional vector space, and let $\textsf{T}∈\mathcal{L}(\textsf{V})$.

I'm trying to define what it means for a basis $\{v_1,\dots,v_n\}$ of $\textsf{V}$ to make $\textsf{T}$ lower triangular.

My thought is let $\dim(\textsf{V})=n$. Then the basis $\{v_1,\dots,v_n\}$ is lower triangular if $\textsf{T}(v_i)∈\operatorname{span}\{v_i,v_{i+1},\dots,v_n\}$ for all $i=1,\dots,n.$

Also, if $\{v_1,\dots,v_n\}$ is a basis for $\textsf{V}$ that makes $\textsf{T}$ upper triangular, then how can I prove that the basis $\{v_n,v_{n−1},\dots,v_1\}$ makes $\textsf{T}$ lower triangular?

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That's one way to describe it, yes.

My first thought was "If the first $k$ components of $v$ are $0$, then the first $k$ components of $Tv$ are zero." That's basically the same thing, though.

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