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My linear algebra textbook makes the following statements:

A vector space is frequently discussed in the text without explicitly mentioning its field of scalars. The reader is cautioned to remember, however, that every vector space is regarded as a vector space over a given field, which is denoted by F.

I understand what a field is (http://mathworld.wolfram.com/Field.html), but I'm wondering what is meant by "over a given field"?

I would greatly appreciate it if someone could please clarify this concept.

Thank you.

user26857
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The Pointer
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  • The scalar multiplication of the vector space $V$ is a map $\mathbb{F} \times V \to V$ is what it means. – Stefan Perko Jan 04 '17 at 08:41
  • This means that the scalars in the scalar multiplication are elements of that field. – Stefan4024 Jan 04 '17 at 08:41
  • @StefanPerko I'm not familiar with a "map". Can you please elaborate using more elementary terms? – The Pointer Jan 04 '17 at 08:44
  • @Stefan4024 So $ab = c$, where $a$, $b$, and $c$ are elements in the field, $F$? – The Pointer Jan 04 '17 at 08:45
  • @StefanPerko function, mapping. If you don't know the definition you should definitely look at up since it is literally used everywhere in all of mathematics. – Stefan Perko Jan 04 '17 at 08:46
  • $b,c$ are elements of the vector space, while $a$ is an element of the field – Stefan4024 Jan 04 '17 at 08:50
  • At the risk of making a confusing statement: Technically, a vector space is really a $4$-tuple $(F,V,+,\cdot)$ where $F$ is a field, $V$ is a set, and $+:V \times V \to V$ and $\cdot: F \times V \to V$ are functions such that certain properties (listed in the definition of a vector space) are satisfied. The tuple $(F,V,+, \cdot)$ is often referred to also as $V$, even though this means that two distinct things now have the same name $V$. – littleO Jan 04 '17 at 09:27

4 Answers4

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It simply means that prior to talking about a vector space, one has to fix the field this will be a vector space over; the axioms of a vector space explicitly refer to elements (scalars) that can be chosen in some given field. Since the text does not want to restrict itself by selecting one particular field (like for instance the real numbers), it supposes a choice of field has been made, but does not say which choice; therefore the chosen field will be designated by the generic identifier$~F$, which could stand for any field (but only one at a time).

For a vague analogy, consider a text about finance. Since it is talking about money, it has to assume a currency is chosen, since monetary transactions require a currency to be expressed in. However, the book has no interest in stating just which currency is chosen, since by say fixing the currency to be £, it would falsely suggest that the general stuff it has to say is only valid in countries that use that currency. Therefore the text might adopt a generic symbol like $C$ to stand for a fixed but unspecified currency

Marc van Leeuwen
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You shouldn't see a vector space as just a set $V$ of vectors $v \in V$, but rather as a structure of:

  • a set $V$ (whose elements we call vectors),
  • together with a field $F$ (whose elements we call scalars),
  • and two operations (called addition and scalar multiplication) that satisfy a number of axioms.

As you probably know, we do not only add vectors $(\vec a + \vec b)$, we also multiply vectors with scalars $(k\vec v)$ and these numbers $k$ have to come from somewhere. They are the elements of the field $F$.

When a field is not explicitly mentioned, it is often assumed that the vector space is taken over the field of the real numbers $\mathbb{R}$. But you can take other fields, such as the rational numbers $\mathbb{Q}$ or the complex numbers $\mathbb{C}$. When relevant, you should mention the field and if you omit it, it's important to still realize it is there.

StackTD
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The field $\mathbb{F}$ is the collection of scalars associated with the vector field. The most common examples of vector spaces are over $\mathbb{R}$ or $\mathbb{C}$, but for many theorems in linear algebra, the only important algebraic properties that the scalars must have are the properties of a field ($\mathbb{R}$ and $\mathbb{C}$ have additional analytic properties).

You use this field in the definition of a vector space when you define scalar multiplication: a map $\mathbb{F} \times V \to V$ which sends $(\alpha,v)$ to the multiplication of $v$ by $\alpha$, written $\alpha v$. Consequently, if you choose a basis, then $V$ can be viewed (in the finite-dimensional case, at least) as column vectors with elements in $\mathbb{F}$, in the same way that an $n$-dimensional real vector space may be viewed as $\mathbb{R}^{n}$ when a basis has been chosen.

For a simple example of using a different field, we can take, say, $\mathbb{F}_{2}^{n}$, which would be all column vectors whose components are only $0$ or $1$.

For a more interesting example: $\mathbb{R}$ can be viewed as an infinite dimensional vector space over the rationals $\mathbb{Q}$.

preferred_anon
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For tangible examples for why it's important, consider $\mathbb C$. This can be viewed as a $1$ dimension vector space over itself, or we can view it as a two dimension vector space over $\mathbb R$, or as an infinite dimension vector space over $\mathbb Q$. These are three perfectly fine fields, but just saying "Let $\mathbb C$ be a vector space" is ambiguous until we know the underlying field.

Mark Schultz-Wu
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