# Definition of applying a polynomial to a linear transformation

Put content here**Definition:** If \(p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k\) is a polynomial and \(T: V 	o V\) is a linear transformation, then \(p(T)\) is the linear transformation:
\[p(T) = a_0 I + a_1 T + a_2 T^2 + \cdots + a_k T^k\]
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where \(T^k\) denotes the \(k\)-fold composition \(T \circ T \circ \cdots \circ T\) and \(I\) is the identity transformation.
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**Example:** If \(p(x) = x^2 - 3x + 2\) and \(T\) is a linear operator, then:
\[p(T) = T^2 - 3T + 2I\]
and for any vector \(\mathbf{v}\): \(p(T)(\mathbf{v}) = T(T(\mathbf{v})) - 3T(\mathbf{v}) + 2\mathbf{v}\).

# Parents

* Eigenvalues and eigenvectors