Nulrummet

Nulrummet, also known as the null space or kernel of a matrix, is a fundamental concept in linear algebra. It refers to the set of all vectors that, when multiplied by a given matrix, yield the zero vector. In mathematical terms, for a matrix *A*, the null space consists of all vectors *x* such that *Ax* = 0. This space is denoted as *N(A)* or *Ker(A)*.

The null space is a subspace of the domain of *A*, meaning it is closed under addition and scalar multiplication. If the null space contains only the zero vector, the matrix is said to be injective (one-to-one). If the null space has more than just the zero vector, the matrix is not injective, indicating that multiple inputs can produce the same output.

The dimension of the null space, known as the nullity of *A*, is related to the rank of the matrix via the rank-nullity theorem. This theorem states that for any matrix *A* mapping from an *m*-dimensional space to an *n*-dimensional space, the rank of *A* plus the nullity of *A* equals the number of columns in *A*. This relationship is expressed as:

rank(*A*) + nullity(*A*) = number of columns in *A*

The null space plays a crucial role in solving systems of linear equations. When a system has infinitely many solutions, the non-trivial solutions form the null space of the coefficient matrix. This concept is also important in applications such as data compression, signal processing, and solving underdetermined systems.

In summary, nulrummet is a key idea in linear algebra that helps analyze the behavior of linear transformations, particularly in determining whether a transformation is invertible and in solving linear systems.