Minimal Polynomial of a Linear Transformation
The minimal polynomial of a linear transformation T from R^(4) to R^(4) being x^(2) x 1 indicates the smallest degree monic polynomial that evaluates to zero when the linear transformation is applied. Let's delve into what this means in detail:

Definition of Minimal Polynomial
- The minimal polynomial of a linear transformation is the monic polynomial of smallest degree that annihilates the transformation, i.e., T satisfies its own minimal polynomial.

Given Minimal Polynomial
- In this case, the minimal polynomial of T is x^(2) x 1. This means that T satisfies the polynomial equation T^(2) + T = 0.

Implications of Minimal Polynomial
- The minimal polynomial provides information about the behavior of the linear transformation T. In this instance, it tells us that T satisfies a polynomial equation of degree 2.

Eigenvalues and Eigenvectors
- The roots of the minimal polynomial correspond to the eigenvalues of the linear transformation. In this case, the eigenvalues can be found by solving the equation x^(2) x 1 = 0.

Matrix Representation
- The minimal polynomial also helps in finding the matrix representation of the linear transformation. By expressing T in terms of its minimal polynomial, one can construct the matrix that represents T.
By understanding the concept of the minimal polynomial and its implications, we can gain insights into the properties and behavior of the given linear transformation T from R^(4) to R^(4) with the minimal polynomial x^(2) x 1.