Editing Kernel (linear algebra) (section)

==Examples==

*If {{math|''L'': '''R'''<sup>''m''</sup> → '''R'''<sup>''n''</sup>}}, then the kernel of {{math|''L''}} is the solution set to a homogeneous [[system of linear equations]].  As in the above illustration, if {{math|''L''}} is the operator: <math display="block"> L(x_1,  x_2, x_3) = (2 x_1 + 3 x_2 + 5 x_3,\; - 4 x_1 + 2 x_2 + 3 x_3)</math> then the kernel of {{math|''L''}} is the set of solutions to the equations <math display="block"> \begin{alignat}{7}
    2x_1 &\;+\;& 3x_2 &\;+\;& 5x_3 &\;=\;& 0 \\
   -4x_1 &\;+\;& 2x_2 &\;+\;& 3x_3 &\;=\;& 0
\end{alignat}</math>
*Let {{math|''C''[0,1]}} denote the [[vector space]] of all continuous real-valued functions on the interval [0,1], and define {{math|''L'': ''C''[0,1] → '''R'''}} by the rule <math display="block">L(f) = f(0.3).</math> Then the kernel of {{math|''L''}} consists of all functions {{math|1=''f'' ∈ ''C''[0,1]}} for which {{math|1=''f''(0.3) = 0}}.
*Let {{math|''C''<sup>∞</sup>('''R''')}} be the vector space of all infinitely differentiable functions {{math|'''R''' → '''R'''}}, and let {{math|''D'': ''C''<sup>∞</sup>('''R''') → ''C''<sup>∞</sup>('''R''')}} be the [[differential operator|differentiation operator]]: <math display="block">D(f) = \frac{df}{dx}.</math> Then the kernel of {{math|''D''}} consists of all functions in {{math|''C''<sup>∞</sup>('''R''')}} whose derivatives are zero, i.e. the set of all [[constant function]]s.
*Let {{math|'''R'''<sup>∞</sup>}} be the [[direct product]] of infinitely many copies of {{math|'''R'''}}, and let {{math|''s'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>}} be the [[shift operator]] <math display="block"> s(x_1, x_2, x_3, x_4, \ldots) = (x_2, x_3, x_4, \ldots).</math> Then the kernel of {{math|''s''}} is the one-dimensional subspace consisting of all vectors {{math|(''x''<sub>1</sub>, 0, 0, 0, ...)}}.
*If {{mvar|V}} is an [[inner product space]] and {{mvar|W}} is a subspace, the kernel of the [[projection (linear algebra)|orthogonal projection]] {{math|''V'' → ''W''}} is the [[orthogonal complement]] to {{mvar|W}} in {{mvar|V}}.