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}}}}. It is indeed isomorphic to ''H''.

Given a [[isomorphism]] <math>\varphi: G \to H</math>, one has <math>\ker \varphi = 1</math>.<ref name=":0" /> 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=":0" /> 

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=":0" /> 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=":2" />

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:
: <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=":0" />

=== 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=":0" />

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=":0" /> If we instead work with polynomials with real coefficients, then we again receive a homomorphism with its kernel being the polynomials with constant term 0.<ref name=":2" />

=== Linear maps ===
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If ''V'' and ''W'' are [[finite-dimensional vector space|finite-dimensional]] and [[basis (linear algebra)|bases]] have been chosen, then ''T'' can be described by a [[matrix (mathematics)|matrix]] ''M'', and the kernel can be computed by solving the homogeneous [[system of linear equations]] {{nowrap|1=''M'''''v''' = '''0'''}}. In this case, the kernel of ''T'' may be identified to the [[kernel (matrix)|kernel of the matrix]] ''M'', also called "null space" of ''M''. The dimension of the null space, called the nullity of ''M'', is given by the number of columns of ''M'' minus the [[rank (matrix theory)|rank]] of ''M'', as a consequence of the [[rank–nullity theorem]].

Solving [[homogeneous differential equation]]s often amounts to computing the kernel of certain [[differential operator]]s.
For instance, in order to find all twice-[[differentiable function]]s ''f'' from the [[real line]] to itself such that
: <math>x f''(x) + 3 f'(x) = f(x),</math>
let ''V'' be the space of all twice differentiable functions, let ''W'' be the space of all functions, and define a linear operator ''T'' from ''V'' to ''W'' by
: <math>(Tf)(x) = x f''(x) + 3 f'(x) - f(x)</math>
for ''f'' in ''V'' and ''x'' an arbitrary [[real number]].
Then all solutions to the differential equation are in {{nowrap|ker ''T''}}.