Editing Kernel (algebra) (section)

== Survey of examples ==

=== Group homomorphisms ===
Let ''G'' be the [[cyclic group]] on 6 elements {{nowrap|{{mset|0, 1, 2, 3, 4, 5}}}} with [[modular arithmetic|modular addition]], ''H'' be the cyclic on 2 elements {{nowrap|{{mset|0, 1}}}} with modular addition, and ''f'' the homomorphism that maps each element ''g'' in ''G'' to the element ''g'' modulo 2 in ''H''. Then {{nowrap|ker ''f'' {{=}} {0, 2, 4} }}, since all these elements are mapped to 0<sub>''H''</sub>. The quotient group {{nowrap|''G'' / (ker ''f'')}} has two elements: {{nowrap|{{mset|0, 2, 4}}}} and {{nowrap|{{mset|1, 3, 5}}}}, and is isomorphic to ''H''.<ref name="Dummit Group Kernel Examples">{{harvnb|Dummit|Foote|2004|pp=78–80}}</ref>

Given a [[isomorphism]] <math>\varphi: G \to H</math>, one has <math>\ker \varphi = 1</math>.<ref name="Dummit Group Kernel Examples" /> On the other hand, if this mapping is merely a homomorphism where ''H'' is the trivial group, then <math>\varphi(g)=1</math> for all <math>g \in G</math>, so thus <math>\ker \varphi = G</math>.<ref name="Dummit Group Kernel Examples" />

Let <math>\varphi: \mathbb{R}^2 \to \mathbb{R}</math> be the map defined as <math>\varphi((x,y)) = x</math>. Then this is a homomorphism with the kernel consisting precisely the points of the form <math>(0,y)</math>. This mapping is considered the "projection onto the x-axis."<ref name="Dummit Group Kernel Examples" /> A similar phenomenon occurs with the mapping <math>f: (\mathbb{R}^\times)^2 \to \mathbb{R}^\times </math> defined as <math>f(a,b)=b</math>, where the kernel is the points of the form <math>(a,1)</math><ref name="Hungerford Kernel"/>

For a non-abelian example, let <math>Q_8</math> denote the [[Quaternion group]], and <math>V_4</math> the [[Klein four-group|Klein 4-group]]. Define a mapping <math>\varphi: Q_8 \to V_4</math> to be:<ref name="Dummit Group Kernel Examples" />
: <math>\varphi(\pm1)=1</math>
: <math>\varphi(\pm i)=a</math>
: <math>\varphi(\pm j)=b</math>
: <math>\varphi(\pm k)=c</math>
Then this mapping is a homomorphism where <math>\ker \varphi = \{ \pm 1 \} </math>.<ref name="Dummit Group Kernel Examples" />

=== Ring homomorphisms ===

Consider the mapping <math> \varphi : \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} </math> where the later ring is the integers modulo 2 and the map sends each number to its [[Parity (mathematics)|parity]]; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.<ref name="Dummit Ring Kernel Examples">{{harvnb|Dummit|Foote|2004|p=240}}</ref>

Let <math> \varphi: \mathbb{Q}[x] \to \mathbb{Q} </math> be defined as <math>\varphi(p(x))=p(0)</math>. This mapping , which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero [[if and only if]] said polynomial's constant term is 0.<ref name="Dummit Ring Kernel Examples" /> Polynomials with real coefficients can receive a similar homomorphism, with its kernel being the polynomials with constant term 0.<ref>{{harvnb|Hungerford|2014|p=155}}</ref>

=== 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" />