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Concept# Kernel (linear algebra)

Summary

In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map L : V → W between two vector spaces V and W, the kernel of L is the vector space of all elements v of V such that L(v) = 0, where 0 denotes the zero vector in W, or more symbolically:
The kernel of L is a linear subspace of the domain V.
In the linear map two elements of V have the same in W if and only if their difference lies in the kernel of L, that is,
From this, it follows that the image of L is isomorphic to the quotient of V by the kernel:
In the case where V is finite-dimensional, this implies the rank–nullity theorem:
where the term refers the dimension of the image of L, while refers to the dimension of the kernel of L,
That is,
so that the rank–nullity theorem can be restated as
When V is an inner product space, the quotient can be identified with the orthogonal complement in V of This is the generalization to linear operators of the row space, or coimage, of a matrix.
Module (mathematics)
The notion of kernel also makes sense for homomorphisms of modules, which are generalizations of vector spaces where the scalars are elements of a ring, rather than a field. The domain of the mapping is a module, with the kernel constituting a submodule. Here, the concepts of rank and nullity do not necessarily apply.
Topological vector space
If V and W are topological vector spaces such that 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.
Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically or ), that is operating on column vectors x with n components over K.
The kernel of this linear map is the set of solutions to the equation Ax = 0, where 0 is understood as the zero vector. The dimension of the kernel of A is called the nullity of A.

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In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.

Row echelon form

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. A matrix being in row echelon form means that Gaussian elimination has operated on the rows, and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. Therefore, only row echelon forms are considered in the remainder of this article. The similar properties of column echelon form are easily deduced by transposing all the matrices.

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