Vector Spaces

Definition.

A vector space over a field $F$, or an $F$-vector space, is an abelian group $V$ furnished with a scalar multiplication

$$F\times V\to V,(\lambda ,v)\mapsto \lambda v$$

and satisfying some axioms:

• $\forall \lambda \in V$, the map $m_{\lambda }:v\mapsto \lambda v$ is a homomorphism form $V$ to $V$ (1)
• $\lambda \mapsto m_{\lambda }$ is a homomorphism of rings from $F$ to the ring of homomorphisms from $V$ to $V$ (2)

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This long-winded definition of a vector space shows that they are essentially fancy abelian groups. The fact that $F$ is an abelian group implies a commutative addition operation for which $V$ is closed under. The additional axioms regarding scalar multiplication simply assert the distributivity (1) and associativity (2) of scalar multiplication. Proving results for $F$-vector spaces without specifying any field $F$ allows us to prove incredibly general results. The most widely used fields are of course the real $\mathbb{R}$ and the complex $\mathbb{C}$.

Mathematical jargon aside, vector spaces are powerful because of the level of abstraction they operate at. Part of the reason linear algebra is so widely applicable is because a very wide range of things can be represented as vectors. Such vector representations in turn form a vector space, the collection of all possible forms of said thing. Any result we derive for the abstract notion of a vector space applies to every instance of a vector space.

One example of a useful vector space from mathematics is the space of all real-valued continuous functions defined on a closed interval. Clearly, this set is closed under addition and scalar multiplication, and the other axioms also hold. Thus, we can apply any linear algebraic result to this set. For example, we can apply the Cauchy Shwartz Inequality which holds for abstract vector spaces in general, to obtain the following highly non-trivial inequality for any two real-valued continuous functions $f(x)$ and $g(x)$ defined on a closed interval $[a,b]$: ${\left({\int }_a^{b}f(t)g(t)dt\right)}^2\le \left({\int }_a^b(f(t){)}^{2}dt\right)\left({\int }_a^b(g(t){)}^{2}dt\right).$

Another example of a useful vector space, this time from computer science, is the space of word embeddings. This vector space forms the backbone of large language models such as ChatGPT. We can use a standard dot product to treat the space of words as a geometric inner product space. Because vectors encode semantic meaning in this space, a dot product returns the semantic similarity between words.

Resources: 3Blue1Brown video on vector spaces