The minimal polynomial of a linear transformation exists and is unique.

Theorem: For every linear transformation (T: V o V) on a finite-dimensional nontrivial vector space, the minimal polynomial exists and is unique.

Existence: The space (\mathcal{L}(V,V)) of linear operators on (V) has dimension (n^2) (where (n = \dim V)). Therefore the (n^2 + 1) operators (I, T, T^2, \ldots, T^{n^2}) are linearly dependent, so some nonzero polynomial (p) satisfies (p(T) = 0). Among all such polynomials, there is one of least degree.

Uniqueness: If two monic polynomials (m_1) and (m_2) both have minimal degree and annihilate (T), then (m_1 - m_2) also annihilates (T) but has lower degree, which is only possible if (m_1 - m_2 = 0), i.e., (m_1 = m_2).

Making the polynomial monic (leading coefficient = 1) ensures uniqueness.