Examples of using Vector space in English and their translations into Turkish
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Every vector space has a basis.
This is the definition of a vector space.
Then the scalars of that vector space will be the elements of the associated field.
There is an analogy with the theory of vector space dimensions.
A vector space equipped with a scalar product is called an inner product space. .
It is important to note, however, that this is a vector space grading only.
The dimension of a vector space is well-defined by the dimension theorem for vector spaces. .
Kismet has an underlying, three-dimensional emotional space, a vector space, of where it is emotionally.
A vector space with such an order is called an ordered vector space. .
In mathematics, a bilinear form on a vector space V is a bilinear map V× V→ K, where K is the field of scalars.
A vector space equipped with such an inner product is known as a(real) inner product space. .
Other algebraic properties====* Any linear combination of even functions is even,and the even functions form a vector space over the reals.
The vector space of all real functions is the direct sum of the subspaces of even and odd functions.
For instance, there exists a basis for the real numbers considered as a vector space over the rationals, but no explicit basis has been constructed.
More generally, a vector space may be defined by using any field instead of real numbers, such as complex numbers.
The space Lp for 0< p< 1 is an F-space:it admits a complete translation-invariant metric with respect to which the vector space operations are continuous.
If a vector space"V" over the real numbers R carries an inner product, then the inner product is a bilinear map.
It turns out that if we accept the axiom of choice, every vector space has a basis; nevertheless, this basis may be unnatural, and indeed, may not even be constructible.
The vector space C×C, the Cartesian product of the complex numbers with themselves, is also a"complex plane" in the sense that it is a two-dimensional vector space whose coordinates are"complex numbers.
Any uniform space(hence any metric space, topological vector space, or topological group) is a Cauchy space; see Cauchy filter for definitions.
In mathematics, the adjoint representation(or adjoint action) of a Lie group Gis a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
A linear operator T on a vector space V is defined to be nilpotent if there is a positive integer k such that Tk 0.
Although the p-unit ball Bnp around the origin in this metric is"concave", the topology defined on Rn by the metric dp is the usual vector space topology of Rn,hence ℓnp is a locally convex topological vector space.
In the tensor algebra T(V) of a vector space V, the operation⊗{\displaystyle\otimes} becomes a normal(internal) binary operation.
Thus the connection∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve::formula_6from the vector space lying over γ("s") to that over γ"t.
Two-dimensional complex vector space, a"complex plane" in the sense that it is a two-dimensional vector space whose coordinates are complex numbers.
Other properties===* We have: :formula_23:where"P" is the'cyclic permutation' matrix, a specific permutation matrix given by::formula_24* The set of formula_2 circulant matrices forms an"n"-dimensional vector space; this can be interpreted as the space of functions on the cyclic group of order"n", formula_26 or equivalently the group ring.
More concretely, the complex conjugate vector space is the same underlying real vector space(same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure J different multiplication by i.
If a vector space V over the real numbers R carries an inner product, then the inner product is a bilinear map V× V→ R. In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V× V→ F. If V is a vector space with dual space V∗, then the application operator, b(f, v) f(v) is a bilinear map from V∗× V to the base field.
A Lie coalgebra structure on a vector space"E" is a linear map formula_12 which is antisymmetric(this means that it satisfies formula_13, where formula_14 is the canonical flip formula_15) and satisfies the so-called"cocycle condition"(also known as the"co-Leibniz rule"): formula_16.