Remark

Replacing X by an element r of R is also referred to as `substituting r for X', or simply `substituting r'. In fact, there are many more things one can substitute for X, usually not leading to a polynomial function in the sense of the definition.

• If you substitute X² for X you get a new polynomial in X. In general, if you replace X by a polynomial p(X), you get a new polynomial in X.
• Substitution of a square matrix A leads to a new square matrix:
  a₀I + a₁A + ··· + a_mA^m.

  Note that we interpret a₀ as a₀I, where I is the identity matrix.