The definition of a subspace gives conflicting answers? [closed]












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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?










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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


















-1












$begingroup$


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?










share|cite|improve this question











$endgroup$



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
















-1












-1








-1





$begingroup$


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?










share|cite|improve this question











$endgroup$




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






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 31 '18 at 14:11









Bernard

119k740113




119k740113










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
















  • 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












1 Answer
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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?






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$endgroup$




















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    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?






    share|cite|improve this answer











    $endgroup$


















      5












      $begingroup$

      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?






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        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?






        share|cite|improve this answer











        $endgroup$



        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?







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 31 '18 at 14:51

























        answered Dec 31 '18 at 14:10









        David C. UllrichDavid C. Ullrich

        59.8k43893




        59.8k43893















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