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eaglesmath15
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1. The question
Let V be a vector space with the ordered basis β={v1, v2,...,vn}. Define v0=0. Then there exists a linear transformation T:V→V such that T(vj) = vj+vj-1 for j=1,2,...,n. Compute [T]β.
[T]γβ = (aij), 1≤i≤m, 1≤j≤n (where m is dimension of γ and n is the dimension of β (I think)
Basically, I know that T(v1)= v1, and T(v2) = v2 + v1, all the way up to T(vn) = vn + vn-1, but I'm not sure how this helps me form the matrix. Also, I know that the matrix is generally the dimension of the range by the dimension of the domain, which would make this matrix nxn, but I'm just not sure how to get it.
Let V be a vector space with the ordered basis β={v1, v2,...,vn}. Define v0=0. Then there exists a linear transformation T:V→V such that T(vj) = vj+vj-1 for j=1,2,...,n. Compute [T]β.
Homework Equations
[T]γβ = (aij), 1≤i≤m, 1≤j≤n (where m is dimension of γ and n is the dimension of β (I think)
The Attempt at a Solution
Basically, I know that T(v1)= v1, and T(v2) = v2 + v1, all the way up to T(vn) = vn + vn-1, but I'm not sure how this helps me form the matrix. Also, I know that the matrix is generally the dimension of the range by the dimension of the domain, which would make this matrix nxn, but I'm just not sure how to get it.