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
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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;
$begingroup$
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
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
$endgroup$
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
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 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
11 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours 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
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
$endgroup$
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
linear-algebra
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
edited 10 hours ago
dmtri
1,8212 gold badges5 silver badges21 bronze badges
1,8212 gold badges5 silver badges21 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
asked 11 hours ago
JOHN JOHN
5111 silver badge10 bronze badges
5111 silver badge10 bronze badges
$begingroup$
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
11 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours ago
add a comment |
$begingroup$
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
11 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours ago
$begingroup$
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
11 hours ago
$begingroup$
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
11 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours ago
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
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.
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours ago
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.
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 bronze badges
$endgroup$
add a comment |
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2 Answers
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active
oldest
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2 Answers
2
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votes
$begingroup$
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.
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours 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.
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours 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.
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 bronze badges
$endgroup$
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.
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 bronze badges
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 bronze badges
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 bronze badges
answered 10 hours ago
Jamie RadcliffeJamie Radcliffe
5263 silver badges5 bronze badges
5263 silver badges5 bronze badges
$begingroup$
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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours ago
add a comment |
$begingroup$
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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours ago
$begingroup$
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
10 hours ago
$begingroup$
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
10 hours 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
10 hours 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
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours ago
$begingroup$
@John: You may want to read about binary operation for context.
$endgroup$
– Shahab
10 hours ago
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.
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 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.
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 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.
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 bronze badges
$endgroup$
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.
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 bronze badges
edited 7 hours ago
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 bronze badges
answered 8 hours ago
CopyPasteItCopyPasteIt
5,3851 gold badge9 silver badges30 bronze badges
answered 8 hours ago
CopyPasteItCopyPasteIt
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5,3851 gold badge9 silver badges30 bronze badges
add a comment |
add a comment |
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$begingroup$
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
11 hours ago
$begingroup$
@HagenvonEitzen: How do you define action of $k$?
$endgroup$
– Shahab
10 hours ago