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As in the case of the "dimension of image" characterization, this can be generalized to a definition of the rank of any linear map: the rank of a linear map f : V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. Unfortunately, this definition does not ...
Constant rank maps have a number of nice properties and are an important concept in differential topology. Three special cases of constant rank maps occur. A constant rank map f : M → N is an immersion if rank f = dim M (i.e. the derivative is everywhere injective), a submersion if rank f = dim N (i.e. the derivative is everywhere surjective),
Rank–nullity theorem. The rank–nullity theorem is a theorem in linear algebra, which asserts: the number of columns of a matrix M is the sum of the rank of M and the nullity of M; and; the dimension of the domain of a linear transformation f is the sum of the rank of f (the dimension of the image of f) and the nullity of f (the dimension of ...
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear map (also called, in some contexts, linear transformation or linear mapping) is a map: that is compatible with addition and scalar multiplication, that is
A linear map : between two topological vector spaces is said to be compact if there exists a neighborhood of the origin in such that () is a relatively compact subset of . [ 3 ] Let X , Y {\displaystyle X,Y} be normed spaces and T : X → Y {\displaystyle T:X\to Y} a linear operator.
In the other direction, if F : V ⊗ V → K is a linear map the corresponding bilinear form is given by composing F with the bilinear map V × V → V ⊗ V that sends (v, w) to v⊗w. The set of all linear maps V ⊗ V → K is the dual space of V ⊗ V , so bilinear forms may be thought of as elements of ( V ⊗ V ) ∗ which (when V is ...
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.