Basis, Linear Transformation, and Powers of a Matrix

[math222]

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.

 

[Mark44]

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.

Consider this matrix
$$ A = \begin{bmatrix} 0 & 1 \\ 0 & 0\end{bmatrix}$$

Here A ≠ 0, but A² = 0. This is an example of a 2 x 2 matrix; it's not too hard to find examples among 3 x 3 matrices.

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.

I don't know that the diagonal is important here. I would start with the definition of a basis, particularly the part about the basis vectors being linearly independent. Since the vectors are in R³, and you have three vectors (v, Av, A²v), there are enough of them to form a basis, as long as they are linearly independent.

 

[math222]

Ya I think I get it now. I was able to prove the basis. I'm doing the transformation now. Thanks for your help.