- #1
gravenewworld
- 1,132
- 26
Suppose T is an element of L(V) and (v1, ..., vn) is a basis of V. Then
-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)
.
.
.
.
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
-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)
.
.
.
.
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