math222
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Homework Statement
Let A be an 3x3 matrix so that A^3 = {3x3 zero matrix}. Assume there is a vector
v with [A^2][v] ≠ {zero vector}.
(a) Prove that B = {v; Av; [A^2]v} is a basis.
(b) Let T be the linear transformation represented by A in the stan-
dard basis. What is [T]B?
Homework Equations
A basis must span the space and be linearly independent. Usually the way we find power matrices is through diagonalization, but I'm not sure how that will happen here.
The Attempt at a Solution
I'm having trouble understanding how the power of a matrix can become the zero matrix. I'm trying to come up with an example of a matrix and can't really think of anything. I think I need to be able to understand what the diagonal means in this case, but I'm not sure.