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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 .
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map S : M × N → X {\\displaystyle S:M\\times N\\to X} is a vector space isomorphism .
When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
Linear subspace, in linear algebra, a subset of a vector space that is closed under addition and scalar multiplication; Flat (geometry), a Euclidean subspace; Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector space
The orthogonal complement is always closed in the metric topology. In finite-dimensional spaces, that is merely an instance of the fact that all subspaces of a vector space are closed. In infinite-dimensional Hilbert spaces, some subspaces are not closed, but all orthogonal
It is homogeneous. An affine space need not be included into a linear space, but is isomorphic to an affine subspace of a linear space. All n-dimensional affine spaces over a given field are mutually isomorphic. In the words of John Baez, "an affine space is a vector space that's forgotten its origin". In particular, every linear space is also ...
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