2007年12月19日 · A vector field on that Euclidean space (usually assumed to be on the whole of Euclidean space unless otherwise stated) is a function that associates a particular vector with each point of Euclidean space. We think of there being a separate vector space at each point, called the tangent space at that point.

2020年6月21日 · It's the old question whether you consider realizations of isomorphic mathematical structures as the same or not. E.g. you can realize a 2D real vector space as arrows on an affine plane, ordered pairs of real numbers, the solutions of a 2nd order ODE, the set of real Fibonacci sequences and for sure many more "things".

A vector space is an algebraic object with its characteristic operations, and an affine space is a group action on a set, specifically a vector space acting on a set faithfully and transitively. Why do we say that the origin is no longer special in the affine space?

2015年3月28日 · A vector space is just a set in which you can add and multiply by elements of the base field. You can add polynomials together and multiply them by real numbers (in a way satisfying the axioms,) so polynomials form a vector space. A vector is nothing more or less than an element of a vector space, so polynomials can be seen as vectors.

A vector space is a set in which you can add its elements with one another and in which you can "stretch" its elements, in other words multiply them by a real number. The elements of the vector space are the vectors. Some people would say vector spaces are sets with "more structure".

2014年7月27日 · Also, there may be several (non-equivalent) ways to build up vector spaces from the same group. For example, the additive group of real numbers, $(\mathbb R, +)$ can be a one-dimensional vector space over the field $\mathbb R$, or an (uncountably) infinite-dimensional vector space over $\mathbb Q$ (the rationals).

And the reason the author can do this is that, as it turns out, every vector space is a free object in the category of vector spaces (at least, every finite dimensional vector space is; the statement that every vector space is a free object, that is that every vector space has a basis, is equivalent to the Axiom of Choice).

2014年3月31日 · Vector space contains the 10 axioms and act under those axioms. To prove some new mathematical operation or set is a vector space, you need to prove all 10 axioms hold with those mathematical operations.

The space of all meromorphic functions over a Riemann surface is a field and a vector space over $\mathbb{C}$. The space of rational function over a field $\mathbb{K}$, noted as $\mathbb{K}(x)$ form a field.

To be a vector space, one needs to specify a set, a field, an addition operation and a scalar product operation. Then one needs to check the axioms. Without that, the question is meaningless. The intention was probably to ask whether $\mathbb Q$ is a vector space over the field $\mathbb Q$ with the ordinary notions of addition and ...