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In linear algebra and functional analysis, the kernel of a linear operator L is the set of all operands v for which L(v) = 0. That is, if L: V → W, then
## Examples

## Properties

If L: V → W, then two elements of V have the same image in W if and only if their difference lies in the kernel of L:
## Kernels in functional analysis

If V and W are topological vector spaces (and W is finite-dimensional) then a linear operator L: V → W is continuous if and only if the kernel of L is a closed subspace of V.
## See also

- $ker(L)\; =\; left\{\; vin\; V\; :\; L(v)=0\; right\}text\{,\}$

The kernel of a linear operator R^{m} → R^{n} is the same as the null space of the corresponding n × m matrix. Sometimes the kernel of a general linear operator is referred to as the null space of the operator.

- If L: R
^{m}→ R^{n}, then the kernel of L is the solution set to a homogeneous system of linear equations. For example, if L is the operator:- $L(x\_1,x\_2,x\_3)\; =\; (2x\_1\; +\; 5x\_2\; -\; 3x\_3,;\; 4x\_1\; +\; 2x\_2\; +\; 7x\_3)$

- $begin\{alignat\}\{7\}$

2x_1 &&; + ;&& 5x_2 &&; - ;&& 3x_3 &&; = ;&& 0

4x_1 &&; + ;&& 2x_2 &&; + ;&& 7x_3 &&; = ;&& 0

end{alignat}text{.} - Let C[0,1] denote the vector space of all continuous real-valued functions on the interval [0,1], and define L: C[0,1] → R by the rule
- $L(f)\; =\; f(0.3)text\{.\},$

- Let C
^{∞}(R) be the vector space of all infinitely differentiable functions R → R, and let D: C^{∞}(R) → C^{∞}(R) be the differentiation operator:- $D(f)\; =\; frac\{df\}\{dx\}text\{.\}$

Then the kernel of D consists of all functions in C

^{∞}(R) whose derivatives are zero, i.e. the set of all constant functions. - Let R
^{∞}be the direct sum of infinitely many copies of R, and let s: R^{∞}→ R^{∞}be the shift operator- $s(x\_1,x\_2,x\_3,x\_4,ldots)\; =\; (x\_2,x\_3,x\_4,ldots)text\{.\}$

_{1}, 0, 0, ...). Note that s is onto, despite having nontrivial kernel. - If V is an inner product space and W is a subspace, the kernel of the orthogonal projection V → W is the orthogonal complement to W in V.

- $L(v)\; =\; L(w);;;;Leftrightarrow;;;;L(v-w)=0text\{.\}$

- $text\{im\}(L)\; cong\; V\; /\; ker(L)text\{.\}$

- $dim(ker\; L)\; +\; dim(text\{im\},L)\; =\; dim(V)text\{.\},$

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Last updated on Tuesday June 03, 2008 at 16:02:59 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday June 03, 2008 at 16:02:59 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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