**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.