This is consistent with the universal mapping property underlying the definition of "free vector space", i.e., every vector space can be viewed as (or more accurately, canonically endowed with the …

When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a

Aug 11, 2020 · A vector space is a set of elements (called "vectors"), along with some form of vector addition and scalar multiplication, subject to a list of requirements for how these two operations behave.

We define what is a vector and what we mean by a vector been generated by other vectors. We say that if a vector is some linear combination of other vectors - with respect to elements of some field (a …

A vector space is a set of elements (called vectors) which is defined "over a field" in the sense that if you multiply by a number in the field (think real numbers), you still get an element in the vector space.

Sep 27, 2021 · In another lecture, a professor said that “over a field” means that the components of the elements in the vector space are from some field F. After hearing their explanations, I got confused – …

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.

Since you are working in a subspace of $\mathbb {R}^2$, which you already know is a vector space, you get quite a few of these axioms for free. Namely, commutativity, associativity and distributivity. With …

Finally, note that $\mathbb {C}$ is a vector space ( of dimension 2) over $\mathbb {R}$ because a complex number $ x+iy$ can be identified with the couple of real numbers $ (x,y) \in \mathbb {R}^2$ …

A mathematical vector space is defined abstractly - I suggest you look up vector spaces online or in a text book, and check the definition. One motivating factor for looking at vector spaces is that …