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In every physical textbook on linear algebra that I own, vector spaces are defined as

  • a set $\mathcal{S}$, along with two operations:
    • (vector) addition $\oplus$, and
    • scalar multiplication $\odot$,

that, together, satisfy ten properties (5 properties of addition, 5 properties of scalar multiplication).

However, the Wikipedia article on Vector Spaces lists only 8 axioms / properties, stating (emphasis added):

Vector addition and scalar multiplication are operations, satisfying the closure property: $\vec{u} + \vec{v}$ and $a\vec{v}$ are in $\mathcal{V}$ for all $a$ in $\mathbb{F}$, and $\vec{u},\, \vec{v}$ in $\mathcal{V}$. Some older sources mention these properties as separate axioms.

This statement seems to suggest that the closure axioms are somehow included in the other 8 axioms. Unfortunately, the reason as to why closure under vector addition and scalar multiplication need not be included is not explained.

Further searches online have turned up lists of 8, 9 or 10 properties of Vectors Spaces, so I am a bit confused as to what's going on, here?

N.B. when defining vector addition and and scalar multiplication (see the end of this post for the complete quote), the Wikipedia article does specify that

the resultant vector is also an element of the set $\mathcal{V}$

so are they basically shifting the "burden" of this property onto the operations, themselves? That is certainly what it seems like, but it is not obvious as to why they would make this move with these specific properties, and not the others. Any clarification would be greatly appreciated!

Complete Definition from Wikipedia:

A vector space over a field ${F}$ is a set $V$ together with two operations that satisfy the eight axioms listed below. In the following, $V × V$ denotes the Cartesian product of $V$ with itself, and → denotes a mapping from one set to another.

  • The first operation, called vector addition or simply addition + : $V × V$ → $V$, takes any two vectors $\mathbf v$ and $\mathbf w$ and assigns to them a third vector which is commonly written as $\mathbf v + \mathbf w$, and called the sum of these two vectors. (The resultant vector is also an element of the set $V$.)
  • The second operation, called scalar multiplication · : $F × V$ → $V$， takes any scalar $a$ and any vector $\mathbf v$ and gives another vector $a \mathbf v$. (Similarly, the vector $a \mathbf v$ is an element of the set $V$ ...)
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    $\begingroup$ Yes, already the notation $F\times V\to V$ and $V\times V\to V$ implicitly assume that the given operation is defined for any two elements, and that the result is a member of $V$. $\endgroup$
    – Berci
    Mar 15, 2020 at 21:19
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    $\begingroup$ @Berci -- thank you very much for taking the time to reply & clarify. $\endgroup$
    – Rax Adaam
    Mar 15, 2020 at 21:31
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    $\begingroup$ The two bullet points in your quotation correspond directly to the two axioms that you feel are missing from. It is very unclear to me what the mathematical content of your question is: there are generally many equivalent ways of axiomatising a class of mathematical structures, so there's no reason to be surprised that different references may not agree on the precise details of an axiomatisation. $\endgroup$
    – Rob Arthan
    Mar 15, 2020 at 21:32
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    $\begingroup$ @RobArthan Yes, I understand that, now. Thank you for the additional comments about axiomatization -- I was aware that differences could arise, but it wasn't obvious to me why only these properties were shifted onto the definition of the operations (now I understand that it is because they are (/ can be) addressed by the way we represent the mapping, itself, whereas the other properties are not). I find it's generally instructive for me, at my level, to ensure that I'm comfortable with, & understand the differences between, equivalent formulations. $\endgroup$
    – Rax Adaam
    Mar 15, 2020 at 21:39
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    $\begingroup$ Your comment does clarify what you were concerned about. Apologies if I (unintentionally) sounded a bit dismissive about your question (which I've just upvoted $\ddot{\smile}$). $\endgroup$
    – Rob Arthan
    Mar 15, 2020 at 21:46

1 Answer 1

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I think the Wikipedia article and all of your linear algebra books (as well as every source that I'm aware of) are all using the same definition of a vector space. I think the confusion stems from how the term "operation" is defined. Sometimes people define operations in such a way that it follows automatically that operations are closed. For example if you define an operation $\oplus$ on a set $S$ to be a map $\oplus:S\times S\to S$, then it follows automatically that $S$ is closed under $\oplus$. So if you're defining operations in such a way that operations are always closed, then you only need eight axioms for a vector space. If you're not assuming that operations are necessarily closed, then you're going to need ten axioms, because everyone's definition of a vector space assumes that the space is closed under vector addition and scalar multiplication.

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    $\begingroup$ I see -- and presumably closure is the only property that can be included in the definition of the operations, in this way. So basically you can define it as a set + 2 operations (with no conditions) + 10 properties that the operations must satisfy, or define it as a set + 2 closed operations + 8 properties that they must satisfy. Six of one or half a dozen of the other... Thanks very much for clarifying. Not having a formal background, I still have the bad habit of scanning over those map statements :\ Very much appreciate your time and knowledge. $\endgroup$
    – Rax Adaam
    Mar 15, 2020 at 21:30
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    $\begingroup$ @RaxAdaam Everything you wrote in the above comment is correct. By the way, I think the question you posted is a good question. Thanks for posting it! $\endgroup$
    – user729424
    Mar 15, 2020 at 21:34

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