Mdthbs

How do you call it when two celestial bodies come as close to each other as they will in their current orbits?

What does it mean to have a subnet mask /32?

Is a butterfly one or two animals?

Church Booleans

How to get the pandadocs from an opportunity?

How big would a Daddy Longlegs Spider need to be to kill an average Human?

Chess software to analyze games

Was 'help' pronounced starting with a vowel sound?

How can I use unicode in this condition?

Why doesn't the Falcon-9 first stage use three legs to land?

The logic of invoking virtual functions is not clear (or it is method hiding?)

Defense against attacks using dictionaries

What professions would a medieval village with a population of 100 need?

What is the hex versus octal timeline?

Why don't politicians push for fossil fuel reduction by pointing out their scarcity?

Is it insecure to have an ansible user with passwordless sudo?

Something in the TV

!I!n!s!e!r!t! !n!b!e!t!w!e!e!n!

To "hit home" in German

How does turbine efficiency compare with internal combustion engines if all the turbine power is converted to mechanical energy?

How to dismiss intrusive questions from a colleague with whom I don't work?

Can you feel passing through the sound barrier in an F-16?

Potential new partner angry about first collaboration - how to answer email to close up this encounter in a graceful manner

Can you be convicted for being a murderer twice?

Do the eight axioms of vector space imply closure?

Does a vector space need to be closed?difference between vector space and subspace of vector spaceHow to determine vector space?Linear Algebra: Vector Space, Standard OperationSubspaces: Does closure under scalar multiplication imply additive identity?Why is the following set not a vector space?Intuitive idea of Vector space of functionA counterexample that shows addition and scalar multiplication is not enough for a vector space?Do matrices always represent vector spaces?Verifying a Vector SpaceHow $C[a, b]$ satisfies the axioms of vector space

.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ margin-bottom:0;
}

3

1

$begingroup$

This post is similar to my question but I do not quite understand the explanation.

Usually, when we try to prove whether a set is a vector space, we will check closure on summation and multiplication, as well as the axioms of vector space.

But, why books like Linear Algebra Done Wrong, when they talk about vector space, they only talk about the 8 axioms?

I am wondering if it is somewhere inside definitions or from the 8 axioms.

linear-algebra

share|cite|improve this question

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually, there is only one axiom for vector spaces. "A vectors space over a field $k$ is an abelian group together with an action of $k$" ;)
  $endgroup$
  – Hagen von Eitzen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @HagenvonEitzen: How do you define action of $k$?
  $endgroup$
  – Shahab
  2 days ago
  

  
  
  
  

add a comment | 

3

1

$begingroup$

This post is similar to my question but I do not quite understand the explanation.

Usually, when we try to prove whether a set is a vector space, we will check closure on summation and multiplication, as well as the axioms of vector space.

But, why books like Linear Algebra Done Wrong, when they talk about vector space, they only talk about the 8 axioms?

I am wondering if it is somewhere inside definitions or from the 8 axioms.

linear-algebra

share|cite|improve this question

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Actually, there is only one axiom for vector spaces. "A vectors space over a field $k$ is an abelian group together with an action of $k$" ;)
  $endgroup$
  – Hagen von Eitzen
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @HagenvonEitzen: How do you define action of $k$?
  $endgroup$
  – Shahab
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

This post is similar to my question but I do not quite understand the explanation.

Usually, when we try to prove whether a set is a vector space, we will check closure on summation and multiplication, as well as the axioms of vector space.

But, why books like Linear Algebra Done Wrong, when they talk about vector space, they only talk about the 8 axioms?

I am wondering if it is somewhere inside definitions or from the 8 axioms.

linear-algebra

share|cite|improve this question

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

$endgroup$

share|cite|improve this question

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

share|cite|improve this question

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

edited 2 days ago

dmtri

1,82122 gold badges55 silver badges2121 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

asked 2 days ago

JOHN JOHN

51311 silver badge1010 bronze badges

2 Answers
2

active

oldest

votes

8

$begingroup$

Treil says "A vector space $V$ is a collection of objects, called vectors..., along with two operations, addition of vectors and multiplication by a number (scalar), such that ...".

He is using operation in sense of functions ${+}: V times Vto V$ and $ cdot: {mathbb F} times V to V$. In particular he is assuming closure as part of the defintion of an operation. When you need to check (for instance) that a subset $W$ of a vector space $V$ is itself a vector space (under the "same" operations), that means you need to check that $+$ and $cdot $ are in fact operations on $W$. This requires that you check $W$ is closed under addition and scalar multiplication.

share|cite|improve this answer

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  for the part $+: V times V mapsto V$, I am not familiar with the notation. Can you explain a bit more? Why $+$ is a function from $V times V$ to $V$?
  $endgroup$
  – JOHN
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Addition is a function that takes an ordered pair as an input, and gives a single element as output. In real numbers, for example, $+(3,5)=8$.
  $endgroup$
  – G Tony Jacobs
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @John: You may want to read about binary operation for context.
  $endgroup$
  – Shahab
  2 days ago
  

  
  
  
  

add a comment | 

1

$begingroup$

The OP should note that we usually denote a function from $V times V$ to $V$ by a letter, say $f$, and write

$$ f: V times V to V$$

and denote the image of $(v,w) in V times V$ under $f$, an element in $V$, by $f(v,w)$.

Now if the operation is addition, $+$, of vectors, we find it both convenient and illuminating to use $vec v + vec w$ as opposed to $+(v,w)$.

Again, we might want to use some letter $f$ to denote a function from $Bbb R times V$ to $V$, so that if $alpha in Bbb R$ and $v in V$ the image under $f$ of $(alpha,v)$ is denoted by $f(alpha,v)$.

But if the operation is scalar multiplication of a vector, $cdot$, we find it both convenient and illuminating to use $alpha cdot vec v$ as opposed to

$$cdot(alpha,v)$$

In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,

$$ alpha vec v$$

Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

$endgroup$

add a comment | 

Your Answer

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3326972%2fdo-the-eight-axioms-of-vector-space-imply-closure%23new-answer', 'question_page');
}
);

Post as a guest

Name

Email

Required, but never shown

8

$begingroup$

Treil says "A vector space $V$ is a collection of objects, called vectors..., along with two operations, addition of vectors and multiplication by a number (scalar), such that ...".

He is using operation in sense of functions ${+}: V times Vto V$ and $ cdot: {mathbb F} times V to V$. In particular he is assuming closure as part of the defintion of an operation. When you need to check (for instance) that a subset $W$ of a vector space $V$ is itself a vector space (under the "same" operations), that means you need to check that $+$ and $cdot $ are in fact operations on $W$. This requires that you check $W$ is closed under addition and scalar multiplication.

share|cite|improve this answer

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  for the part $+: V times V mapsto V$, I am not familiar with the notation. Can you explain a bit more? Why $+$ is a function from $V times V$ to $V$?
  $endgroup$
  – JOHN
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Addition is a function that takes an ordered pair as an input, and gives a single element as output. In real numbers, for example, $+(3,5)=8$.
  $endgroup$
  – G Tony Jacobs
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @John: You may want to read about binary operation for context.
  $endgroup$
  – Shahab
  2 days ago
  

  
  
  
  

add a comment | 

8

$begingroup$

Treil says "A vector space $V$ is a collection of objects, called vectors..., along with two operations, addition of vectors and multiplication by a number (scalar), such that ...".

He is using operation in sense of functions ${+}: V times Vto V$ and $ cdot: {mathbb F} times V to V$. In particular he is assuming closure as part of the defintion of an operation. When you need to check (for instance) that a subset $W$ of a vector space $V$ is itself a vector space (under the "same" operations), that means you need to check that $+$ and $cdot $ are in fact operations on $W$. This requires that you check $W$ is closed under addition and scalar multiplication.

share|cite|improve this answer

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  for the part $+: V times V mapsto V$, I am not familiar with the notation. Can you explain a bit more? Why $+$ is a function from $V times V$ to $V$?
  $endgroup$
  – JOHN
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Addition is a function that takes an ordered pair as an input, and gives a single element as output. In real numbers, for example, $+(3,5)=8$.
  $endgroup$
  – G Tony Jacobs
  2 days ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @John: You may want to read about binary operation for context.
  $endgroup$
  – Shahab
  2 days ago
  

  
  
  
  

add a comment | 

$begingroup$

Treil says "A vector space $V$ is a collection of objects, called vectors..., along with two operations, addition of vectors and multiplication by a number (scalar), such that ...".

He is using operation in sense of functions ${+}: V times Vto V$ and $ cdot: {mathbb F} times V to V$. In particular he is assuming closure as part of the defintion of an operation. When you need to check (for instance) that a subset $W$ of a vector space $V$ is itself a vector space (under the "same" operations), that means you need to check that $+$ and $cdot $ are in fact operations on $W$. This requires that you check $W$ is closed under addition and scalar multiplication.

share|cite|improve this answer

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

$endgroup$

share|cite|improve this answer

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

answered 2 days ago

Jamie RadcliffeJamie Radcliffe

52633 silver badges55 bronze badges

1

$begingroup$

The OP should note that we usually denote a function from $V times V$ to $V$ by a letter, say $f$, and write

$$ f: V times V to V$$

and denote the image of $(v,w) in V times V$ under $f$, an element in $V$, by $f(v,w)$.

Now if the operation is addition, $+$, of vectors, we find it both convenient and illuminating to use $vec v + vec w$ as opposed to $+(v,w)$.

Again, we might want to use some letter $f$ to denote a function from $Bbb R times V$ to $V$, so that if $alpha in Bbb R$ and $v in V$ the image under $f$ of $(alpha,v)$ is denoted by $f(alpha,v)$.

But if the operation is scalar multiplication of a vector, $cdot$, we find it both convenient and illuminating to use $alpha cdot vec v$ as opposed to

$$cdot(alpha,v)$$

In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,

$$ alpha vec v$$

Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

$endgroup$

add a comment | 

1

$begingroup$

The OP should note that we usually denote a function from $V times V$ to $V$ by a letter, say $f$, and write

$$ f: V times V to V$$

and denote the image of $(v,w) in V times V$ under $f$, an element in $V$, by $f(v,w)$.

Now if the operation is addition, $+$, of vectors, we find it both convenient and illuminating to use $vec v + vec w$ as opposed to $+(v,w)$.

Again, we might want to use some letter $f$ to denote a function from $Bbb R times V$ to $V$, so that if $alpha in Bbb R$ and $v in V$ the image under $f$ of $(alpha,v)$ is denoted by $f(alpha,v)$.

But if the operation is scalar multiplication of a vector, $cdot$, we find it both convenient and illuminating to use $alpha cdot vec v$ as opposed to

$$cdot(alpha,v)$$

In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,

$$ alpha vec v$$

Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

$endgroup$

add a comment | 

$begingroup$

The OP should note that we usually denote a function from $V times V$ to $V$ by a letter, say $f$, and write

$$ f: V times V to V$$

and denote the image of $(v,w) in V times V$ under $f$, an element in $V$, by $f(v,w)$.

Now if the operation is addition, $+$, of vectors, we find it both convenient and illuminating to use $vec v + vec w$ as opposed to $+(v,w)$.

Again, we might want to use some letter $f$ to denote a function from $Bbb R times V$ to $V$, so that if $alpha in Bbb R$ and $v in V$ the image under $f$ of $(alpha,v)$ is denoted by $f(alpha,v)$.

But if the operation is scalar multiplication of a vector, $cdot$, we find it both convenient and illuminating to use $alpha cdot vec v$ as opposed to

$$cdot(alpha,v)$$

In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,

$$ alpha vec v$$

Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

$endgroup$

share|cite|improve this answer

edited 2 days ago

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges

answered 2 days ago

CopyPasteItCopyPasteIt

5,39011 gold badge99 silver badges3030 bronze badges