Minimal Polynomial of a Linear Map
In this scenario, the minimal polynomial of the linear map T: R⁵ -> R⁵ is given as x²(x³-1). Let's break down what this means and its implications.

Understanding the Minimal Polynomial
- The minimal polynomial of a linear map is the monic polynomial of least degree that annihilates the linear map. In other words, it is the polynomial of smallest degree that the linear map satisfies.

Given Polynomial x²(x³-1)
- The given minimal polynomial x²(x³-1) suggests that the linear map T must satisfy this polynomial.
- The factors x² and (x³-1) indicate that the linear map T satisfies the conditions imposed by these factors.

Implications
- The factor x² implies that the linear map T satisfies the polynomial x² = 0, which means T² = 0 (the square of the linear map is the zero map).
- The factor (x³-1) implies that the linear map T satisfies the polynomial x³-1 = 0, which means T³ = I (the cube of the linear map is the identity map).

Conclusion
- In conclusion, the minimal polynomial x²(x³-1) indicates that the linear map T satisfies the conditions imposed by the factors x² and (x³-1). This implies that T² = 0 and T³ = I, where I is the identity map.