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

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 …

Oct 2, 2017 · A vector space needs to contain $\vec 0$. Thus any subset of a vector space that doesn't, like $\Bbb R^2 \setminus \ {\vec 0\}\subseteq \Bbb R^2$ with the standard vector operations is not 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.

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.

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.

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$ …

Sep 21, 2019 · Part I. I'm going to give a meta-answer here: most definitions in elementary mathematics (let's say "up to junior year in college") are pretty well established and tested. There are some bad …

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 …

A basis of the vector space V is a subset of linearly independent vectors that span the whole of V. If S = {x1, …, xn} this means that for any vector u ∈ V, there exists a unique system of coefficients such …