RangNullitätssatz

The Rang-Nullitätssatz, also known as the Rank-Nullity Theorem, is a fundamental result in linear algebra that relates the dimensions of the kernel and the image of a linear transformation. It is a direct consequence of the dimension theorem and provides a powerful tool for understanding the behavior of linear maps.

The theorem states that for any linear transformation T from a vector space V to a vector space W, the sum of the dimension of the kernel (null space) of T and the dimension of the image (range) of T is equal to the dimension of the domain V. Mathematically, this can be expressed as:

dim(ker(T)) + dim(im(T)) = dim(V)

where ker(T) denotes the kernel of T, im(T) denotes the image of T, and dim denotes the dimension of a vector space.

The Rang-Nullitätssatz has several important implications. It allows for the determination of the rank and nullity of a linear transformation without explicitly computing the kernel and image. It also provides a criterion for the injectivity and surjectivity of a linear transformation: T is injective if and only if its nullity is zero, and T is surjective if and only if its rank is equal to the dimension of W.

The theorem is widely used in various areas of mathematics and its applications, including the study of linear systems, matrix theory, and functional analysis. Its simplicity and elegance make it a cornerstone of linear algebra and a valuable tool for mathematicians and researchers.