The definition of a subspace gives conflicting answers? [closed]
$begingroup$
In my linear algebra book the definition of a subspace is given as such.
A non-empty set of vectors in $Bbb R^n$ is called a subspace if it is closed under scalar multiplication and addition.
However, this the same book uses the term subspace when defining row space. This leads to conflicting logic.
A row space is the subspace of $Bbb R^n$ that is spanned by the row vectors of a.
If a row space is subspace, that a non-empty set of vectors with certain properties, then how can it be a span at the same time.
What is the precise definition of a subspace and row space?
linear-algebra
edited Dec 31 '18 at 14:11
Bernard
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
$endgroup$
closed as off-topic by RRL, clathratus, Leucippus, Cesareo, ncmathsadist Jan 1 at 0:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, clathratus, Leucippus, Cesareo, ncmathsadist
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
In my linear algebra book the definition of a subspace is given as such.
A non-empty set of vectors in $Bbb R^n$ is called a subspace if it is closed under scalar multiplication and addition.
However, this the same book uses the term subspace when defining row space. This leads to conflicting logic.
A row space is the subspace of $Bbb R^n$ that is spanned by the row vectors of a.
If a row space is subspace, that a non-empty set of vectors with certain properties, then how can it be a span at the same time.
What is the precise definition of a subspace and row space?
linear-algebra
edited Dec 31 '18 at 14:11
Bernard
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
$endgroup$
closed as off-topic by RRL, clathratus, Leucippus, Cesareo, ncmathsadist Jan 1 at 0:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, clathratus, Leucippus, Cesareo, ncmathsadist
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
1
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09
add a comment |
$begingroup$
In my linear algebra book the definition of a subspace is given as such.
A non-empty set of vectors in $Bbb R^n$ is called a subspace if it is closed under scalar multiplication and addition.
However, this the same book uses the term subspace when defining row space. This leads to conflicting logic.
A row space is the subspace of $Bbb R^n$ that is spanned by the row vectors of a.
If a row space is subspace, that a non-empty set of vectors with certain properties, then how can it be a span at the same time.
What is the precise definition of a subspace and row space?
linear-algebra
edited Dec 31 '18 at 14:11
Bernard
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
$endgroup$
In my linear algebra book the definition of a subspace is given as such.
A non-empty set of vectors in $Bbb R^n$ is called a subspace if it is closed under scalar multiplication and addition.
However, this the same book uses the term subspace when defining row space. This leads to conflicting logic.
A row space is the subspace of $Bbb R^n$ that is spanned by the row vectors of a.
If a row space is subspace, that a non-empty set of vectors with certain properties, then how can it be a span at the same time.
What is the precise definition of a subspace and row space?
linear-algebra
linear-algebra
edited Dec 31 '18 at 14:11
Bernard
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
edited Dec 31 '18 at 14:11
Bernard
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
edited Dec 31 '18 at 14:11
Bernard
119k740113
edited Dec 31 '18 at 14:11
Bernard
119k740113
edited Dec 31 '18 at 14:11
Bernard
119k740113
119k740113
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
asked Dec 31 '18 at 14:02
Hung TrinhHung Trinh
283
283
closed as off-topic by RRL, clathratus, Leucippus, Cesareo, ncmathsadist Jan 1 at 0:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, clathratus, Leucippus, Cesareo, ncmathsadist
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by RRL, clathratus, Leucippus, Cesareo, ncmathsadist Jan 1 at 0:49
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – RRL, clathratus, Leucippus, Cesareo, ncmathsadist
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
1
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09
add a comment |
2
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
1
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09
2
2
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
1
1
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It's hard to see what sort of "conflict" you see here.
A subspace of $Bbb R^n$ is defined as you say. Given vectors $v_1,dots, v_kinBbb R^n$, the span of $v_1,dots,v_k$ is the set of linear combinations of $v_1,dots,v_k$. It's an easy theorem that the span of $v_1,dots,v_k$ is a subspace of $Bbb R^n$. Another term for "the span of $v_1,dots,v_k$" is "the subspace of $Bbb R^n$ spanned by $v_1,dots,v_k$". The rowspace of an $mtimes n$ matrix $A$ is the span of the rows of $A$. Hence the rowpsace of $A$ is a subspace of $Bbb R^n$.
Those are the precise definitions - what's the problem?
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
$endgroup$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It's hard to see what sort of "conflict" you see here.
A subspace of $Bbb R^n$ is defined as you say. Given vectors $v_1,dots, v_kinBbb R^n$, the span of $v_1,dots,v_k$ is the set of linear combinations of $v_1,dots,v_k$. It's an easy theorem that the span of $v_1,dots,v_k$ is a subspace of $Bbb R^n$. Another term for "the span of $v_1,dots,v_k$" is "the subspace of $Bbb R^n$ spanned by $v_1,dots,v_k$". The rowspace of an $mtimes n$ matrix $A$ is the span of the rows of $A$. Hence the rowpsace of $A$ is a subspace of $Bbb R^n$.
Those are the precise definitions - what's the problem?
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
$endgroup$
add a comment |
$begingroup$
It's hard to see what sort of "conflict" you see here.
A subspace of $Bbb R^n$ is defined as you say. Given vectors $v_1,dots, v_kinBbb R^n$, the span of $v_1,dots,v_k$ is the set of linear combinations of $v_1,dots,v_k$. It's an easy theorem that the span of $v_1,dots,v_k$ is a subspace of $Bbb R^n$. Another term for "the span of $v_1,dots,v_k$" is "the subspace of $Bbb R^n$ spanned by $v_1,dots,v_k$". The rowspace of an $mtimes n$ matrix $A$ is the span of the rows of $A$. Hence the rowpsace of $A$ is a subspace of $Bbb R^n$.
Those are the precise definitions - what's the problem?
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
$endgroup$
add a comment |
$begingroup$
It's hard to see what sort of "conflict" you see here.
A subspace of $Bbb R^n$ is defined as you say. Given vectors $v_1,dots, v_kinBbb R^n$, the span of $v_1,dots,v_k$ is the set of linear combinations of $v_1,dots,v_k$. It's an easy theorem that the span of $v_1,dots,v_k$ is a subspace of $Bbb R^n$. Another term for "the span of $v_1,dots,v_k$" is "the subspace of $Bbb R^n$ spanned by $v_1,dots,v_k$". The rowspace of an $mtimes n$ matrix $A$ is the span of the rows of $A$. Hence the rowpsace of $A$ is a subspace of $Bbb R^n$.
Those are the precise definitions - what's the problem?
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
$endgroup$
It's hard to see what sort of "conflict" you see here.
A subspace of $Bbb R^n$ is defined as you say. Given vectors $v_1,dots, v_kinBbb R^n$, the span of $v_1,dots,v_k$ is the set of linear combinations of $v_1,dots,v_k$. It's an easy theorem that the span of $v_1,dots,v_k$ is a subspace of $Bbb R^n$. Another term for "the span of $v_1,dots,v_k$" is "the subspace of $Bbb R^n$ spanned by $v_1,dots,v_k$". The rowspace of an $mtimes n$ matrix $A$ is the span of the rows of $A$. Hence the rowpsace of $A$ is a subspace of $Bbb R^n$.
Those are the precise definitions - what's the problem?
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
edited Dec 31 '18 at 14:51
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
answered Dec 31 '18 at 14:10
David C. UllrichDavid C. Ullrich
59.8k43893
59.8k43893
add a comment |
add a comment |
2
$begingroup$
Spans are subspaces and a row space is a span, so...
$endgroup$
– Randall
Dec 31 '18 at 14:06
1
$begingroup$
What is $a$? Anyway, the row space attached to a matrix is a particular example of a subspace. Where do you see a conflict?
$endgroup$
– lulu
Dec 31 '18 at 14:09