Each vector space must have a normal basis for proper analysis.
Each vector space has its own unique properties and operations defined on its vectors.
Furthermore, every vector space is isomorphic to one of this form.
Thus one sees that a vector space need not be isomorphic to its dual if it is infinite dimensional, in contrast to the finite dimensional case.
Stąd widać, że przestrzeń liniowa nie musi być izomorficzna do swojej przestrzeni sprzężonej, jeśli jest ona nieskończeniewymiarowa, w przeciwieństwie do przypadku skończeniewymiarowego.
The dimension of this vector space is called the degree of the extension.
The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented.
Orientacja rzeczywistej przestrzeni liniowej to podział baz uporządkowanych na "dodatnio" zorientowane i "ujemnie" zorientowane.
Every vector is contained within some subspace of the larger vector space.
The professor explained how a linear combination can span a vector space.
The concept of a vector space simplifies many problems in physics and engineering.
A linear transformation can convert one vector space to another without distortion.
In calculus, you frequently employ the principles of vector space in optimization.
To classify a vector space, you must identify its vectors and operations.
A vector space can contain an infinite number of vectors depending on the basis.
Przestrzeń wektorowa może zawierać nieskończoną liczbę wektorów, w zależności od bazy.