Rangnullitás

Rangnullitás, also known as rank-nullity theorem, is a fundamental concept in linear algebra, which relates the dimensions of the kernel and the image of a linear transformation. 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. This can be expressed mathematically as:

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

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

The rank-nullity theorem is a direct consequence of the fundamental theorem of linear maps and is often used to determine the rank and nullity of a linear transformation. It is also useful in understanding the behavior of linear transformations and their associated matrices. The theorem is applicable to both finite-dimensional and infinite-dimensional vector spaces, although the proof may vary depending on the context.