what is the relationship between vector spaces and rings?
2013年11月12日 · A vector space is a a quadruple $(V,\mathbb{F}, +, .)$ consisting of an abelian group $(V, +)$ and field $\mathbb{F}$ acting on it by scalar multiplication $.: \mathbb{F}\times V \to V.$ This scalar multiplication operation is compatible with vector addition and field addition. A ring is a triple $(R, +,\times),$ where $(R, +)$ is an abelian group.
What is the main difference between a vector space and a field?
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.
Why is the set of cosines and sines a vector space? - Physics Forums
2015年8月18日 · Cosines and sines are a vector space because they satisfy all of the properties required for a set to be considered a vector space. They are closed under addition and scalar multiplication, they have an identity element (1 for cosines and 0 for sines), and they have inverses (cosine and sine functions).
What "is" a matrix in the context of a vector space?
The action of matrices on a vector space via matrix multiplication is in some sense the same as the action of linear maps on that vector space. The equivalence depends on choosing bases. $\endgroup$ – James Staff
What are differences between affine space and vector space?
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?
Vector space for solutions of differential equations - Physics Forums
2020年6月17日 · A vector space for solutions of differential equations is different from a regular vector space because the vectors in this space represent solutions to a specific mathematical equation, whereas in a regular vector space, the vectors can …
linear algebra - Why $\ {\mathbf {0}\}$ has dimension zero ...
$\begingroup$ The zero vector itself does not have a dimension. The vector space consisting of only the zero vector has dimension 0. This is because a basis for that vector space is the empty set, and the dimension of a vector space is the cardinality …
Double Dual of Vector Space: Is V** Always Same as V? - Physics …
2020年6月21日 · The double dual of a vector space, denoted by V**, is the space of all linear functionals on the dual space V*. In other words, it is the set of all linear maps from V* to the field of scalars. Is the double dual always the same as the original vector space? Yes, the double dual of a vector space is always isomorphic to the original vector space.
Difference between a vector space and a field? - Physics Forums
2010年9月4日 · FAQ: Difference between a vector space and a field? What is a vector space? A vector space is a mathematical structure that consists of a set of objects, called vectors, that can be added together and multiplied by scalars (usually numbers) to create new vectors. It is often used to model physical quantities such as force, velocity, and ...
Why is the dimension of the vector space , 0? - Physics Forums
2010年4月22日 · This also means that the space is not very interesting from a linear algebra perspective, as there are no non-trivial operations that can be performed on a 0-dimensional vector space. How is the dimension of a vector space determined? The dimension of a vector space can be determined by finding a basis for the space. A basis is a set of ...