Suppose T is an element of L(V) and (v1, ......, vn) is a basis of V. Then(adsbygoogle = window.adsbygoogle || []).push({});

-the matrix of T with respect to (v1,...,vn) is upper triangular

-Tvk is an element of span(v1,....,vk) for each k=1,....,n

-span(v1,...,vk) is invariant under T for each k=1,....,n.

can some please explain why you will get an upper triangular matrix. The book doesn't show why at all because it says it is "obvious," but I just don't see why at all. Maybe I am thinking too hard. It gives a proof of the 2nd and 3rd lines by saying

Tv1 is an element of span(v1)

Tv2 is an element of span(v1,v2)

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Tvk is an element of span (v1,...vK).

What i don't understand is why Tv1 is the element of just span of (v1) and Tv2 is in span(v1,v2) etc. I don't understand why you don't have to consider the entire span of the basis vector for each Tvk

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# Some clarification on upper triangular matrices please.

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