Editing Kernel (algebra) (section)

=== Linear maps ===
Let <math>\varphi: \mathbb{C}^3 \to \mathbb{C}</math> be defined as <math>\varphi(x,y,z) = x+2y+3z</math>, then the kernel of <math>\varphi</math> (that is, the null space) will be the set of points <math>(x,y,z) \in \mathbb{C}^3</math> such that <math>x+2y+3z=0</math>, and this set is a subspace of <math>\mathbb{C}^3</math> (the same is true for every kernel of a linear map).<ref name="Axler Kernel Examples">{{harvnb|Axler|p=59}}</ref>

If <math>D</math> represents the [[derivative]] operator on real [[polynomial]]s, then the kernel of <math>D</math> will consist of the polynomials with deterivative equal to 0, that is the [[constant function]]s.<ref name="Axler Kernel Examples" />

Consider the mapping <math>(Tp)(x)=x^2p(x)</math>, where <math>p</math> is a polynomial with real coefficients. Then <math>T</math> is a linear map whose kernel is precisely 0, since it is the only polynomial to satisfy <math>x^2p(x) = 0</math> for all <math>x \in \mathbb{R}</math>.<ref name="Axler Kernel Examples" />