Question

The question is posted below

Answer 1

Let \[A _{t} = \left[\begin{matrix}2 & t \\ 0 & 2\end{matrix}\right]\]

Are the elements I, \[A _{t}\, A^2_{t}, A^3 _{t}\] linearly independent for all two by two matrices, denoted Mat(2, 2)? What is the dimension of the subspace \[V _{t}\subset \] Mat(2, 2) which they span?

Answer 2

Would I just compute the different powers of At and then have a system of linear equations where each equation equals an entry in the Mat(2, 2) (hence those matrices are in the span of Mat (2,2)?