MathAlgebraLinear algebraLinear operators

Actions with linear operators

Putting all the operations together

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Since linear operators are vectors, they can form linear combinations. But since they also have powers, you can create linear combinations of the powers of the same operator: $p(T) = \lambda_3 T^3 + \lambda_2 T^2 + \lambda_1 T + \lambda_0 I$

These expressions are called polynomials, and they are one of the reasons linear operators are so important.

What is the value of the polynomial $p(T)$ in an arbitrary vector $v$ ?

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p(T) (v) = \lambda_3 T(v) + \lambda_2 T(v) + \lambda_1 T(v) + \lambda_0 v

p(T) (v) = \lambda_3 T^3(v) + \lambda_2 T^2(v) + \lambda_1 T(v) + \lambda_0 T(v)

p(T) (v) = \lambda_3 T^3(v) + \lambda_2 T^2(v) + \lambda_1 T(v) + \lambda_0 v

p(T) (v) = \lambda_3 T(v^3) + \lambda_2 T(v^2) + \lambda_1 T(v) + \lambda_0 v^0

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