![]() ![]() There exists $0\in\F$ such that $0 a=a$ for every $a\in\F$.įor every $a\in\F$, there exists $b\in\F$ for which $a b=0$.įor all $a,b\in\F$, we have that $a b=b a$.įor all $a,b,c\in\F$, we have that $(a b) c=a (b c)$. A field is a set $\F$ of elements called scalars, together with two binary operations: These scalars will be the elements of a field, which we have encountered before in my posts on constructing the rational numbers, but I will give the definition again because it is so important.ĭefinition. Since in general there is no concept of vector multiplication, we will need to bring in additional elements by which we are allowed to multiply our vectors to achieve a scaling effect. We would like our definition to include some way to scale vectors so that we can expand of shrink them in magnitude while preserving their direction. We will make a more abstract and inclusive definition of vector spaces, which are the main objects of study in linear algebra. ![]() However, such arrows are not the only mathematical objects that can be added and scaled, so it would be silly to restrict our attention only to them. This is the sort of vector encountered in introductory physics classes. You have likely encountered the idea of a vector before as some sort of arrow, anchored to the origin in euclidean space with some well-defined magnitude and direction. Vector spaces are defined in a similar manner.Ī vector space is a special kind of set containing elements called vectors, which can be added together and scaled in all the ways one would generally expect. ![]() A topological space is a set and a collection of "open sets" which include the set itself, the empty set, finite intersections and arbitrary unions of open sets. Recall that a group is a set with a binary operation, an identity and inverses for all its elements. Vector spaces and free abelian groups have a lot in common, but vector spaces are more familiar, more ubiquitous and easier to compute with.Īt their core, vector spaces are very simple and their definition will closely mimic that of groups and topological spaces. It's almost ridiculous that I would expose you to free abelian groups before talking about vector spaces and linear algebra. ![]()
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