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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, = () =.
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map.
Linear algebra is the branch of mathematics concerning linear equations such as: ... of 0 (called kernel or null space), are linear subspaces of W and V, ...
The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution ... Introduction to Linear Algebra. American Mathematical Society.
When : is a linear transformation between two finite-dimensional subspaces, with = and = (so can be represented by an matrix ), the rank–nullity theorem asserts that if has rank , then is the dimension of the null space of , which represents the kernel of .
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace [1] [note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces .
Jan 19, 2025; Atlanta, GA, USA; The College Football Playoff National Championship trophy at a press conference at The Westin Peachtree Plaza, Savannah Ballroom.
The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V /ker( T ) is isomorphic to the image of V in W . An immediate corollary , for finite-dimensional spaces, is the rank–nullity theorem : the dimension of V is equal to the dimension of the kernel (the nullity of T ) plus the dimension ...