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Kernel and image of a linear map L from V to W. The kernel of L is a linear subspace of the domain V. [3] [2] In the linear map :, two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is, = () =.
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 the kernel of f). [1 ...
These can be interpreted thus: given a linear equation f(v) = w to solve, the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of degrees of freedom in the space of solutions, if it is not empty;
The kernel of a matrix, also called the null space, is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the ...
In linear algebra and functional analysis, a projection is a linear transformation from a vector space to itself (an endomorphism) such that =. That is, whenever P {\displaystyle P} is applied twice to any vector, it gives the same result as if it were applied once (i.e. P {\displaystyle P} is idempotent ).
The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f . Cokernels are dual to the kernels of category theory , hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel ...
is a K-linear transformation of this vector space into itself. The trace, Tr L/K (α), is defined as the trace (in the linear algebra sense) of this linear transformation. [1] For α in L, let σ 1 (α), ..., σ n (α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
Let End(V) be the set of all linear operators on V. Then Lat(End(V))={0,V}. Given a representation of a group G on a vector space V, we have a linear transformation T(g) : V → V for every element g of G. If a subspace W of V is invariant with respect to all these transformations, then it is a subrepresentation and the group G acts on W in a