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For example, the complex numbers C form a two-dimensional vector space over the real numbers R. Likewise, the real numbers R form a vector space over the rational numbers Q which has (uncountably) infinite dimension, if a Hamel basis exists. [b] If V is a vector space over F it may also be regarded as vector space over K. The dimensions are ...
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.
The example of complex numbers is essentially the same as (that is, it is isomorphic to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number x + i y as representing the ordered pair (x, y) in the complex plane then we see that the rules for addition and scalar multiplication correspond exactly to ...
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
For example, the complex plane treated as a one-dimensional complex linear space may be downgraded to a two-dimensional real linear space. In contrast, the real line can be treated as a one-dimensional real linear space but not a complex linear space. See also field extensions. More generally, a vector space over a field also has the structure ...
A subspace is Lagrangian if and only if it is both isotropic and coisotropic. In a finite-dimensional vector space, a Lagrangian subspace is an isotropic one whose dimension is half that of V. Every isotropic subspace can be extended to a Lagrangian one. Referring to the canonical vector space R 2n above, the subspace spanned by {x 1, y 1} is ...
In mathematics, an invariant subspace of a linear mapping T : V → V i.e. from some vector space V to itself, is a subspace W of V that is preserved by T. More generally, an invariant subspace for a collection of linear mappings is a subspace preserved by each mapping individually.
The fundamental example of a linear complex structure is the structure on R 2n coming from the complex structure on C n.That is, the complex n-dimensional space C n is also a real 2n-dimensional space – using the same vector addition and real scalar multiplication – while multiplication by the complex number i is not only a complex linear transform of the space, thought of as a complex ...