Kernel (linear algebra)
||It has been suggested that Kernel (matrix) be merged into this article. (Discuss) Proposed since November 2012.|
The kernel of a linear map Rm → Rn, or, more generally of a linear map between finite dimensional spaces equipped with bases is the same as the null space of the corresponding n × m matrix. Sometimes the kernel of a linear map is referred to as the null space of the map, and the dimension of the kernel is referred to as the map's nullity.
- If L: Rm → Rn, then the kernel of L is the solution set to a homogeneous system of linear equations. For example, if L is the operator:
- then the kernel of L is the set of solutions to the equations
- 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
- Then the kernel of L consists of all functions f ∈ C[0,1] for which f(0.3) = 0.
- 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:
- 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 product of infinitely many copies of R, and let s: R∞ → R∞ be the shift operator
- Then the kernel of s is the one-dimensional subspace consisting of all vectors (x1, 0, 0, ...).
- 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.
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:
This implies the rank-nullity theorem: