zero vector in and out of a span of vectors

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Is span of the vectors $v_1, v_2, 0$ equivalent to the span of $v_1, v_2$ ?




Im struggling to think whether this statement is true or not mainly because my train of thought is:



all vectors in a span can be multiplied by $0$ to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




linear-algebra vectors




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  • Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
    – David C. Ullrich
    Aug 27 at 14:52















up vote
4
down vote

favorite













Is span of the vectors $v_1, v_2, 0$ equivalent to the span of $v_1, v_2$ ?




Im struggling to think whether this statement is true or not mainly because my train of thought is:



all vectors in a span can be multiplied by $0$ to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




linear-algebra vectors




share|cite|improve this question






















  • Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
    – David C. Ullrich
    Aug 27 at 14:52













up vote
4
down vote

favorite









up vote
4
down vote

favorite












Is span of the vectors $v_1, v_2, 0$ equivalent to the span of $v_1, v_2$ ?




Im struggling to think whether this statement is true or not mainly because my train of thought is:



all vectors in a span can be multiplied by $0$ to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




linear-algebra vectors




share|cite|improve this question















Is span of the vectors $v_1, v_2, 0$ equivalent to the span of $v_1, v_2$ ?




Im struggling to think whether this statement is true or not mainly because my train of thought is:



all vectors in a span can be multiplied by $0$ to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?





linear-algebra vectors





share|cite|improve this question













share|cite|improve this question




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edited Aug 27 at 6:47









LDM

730314




730314










asked Aug 27 at 6:15









lohboys

617




617











  • Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
    – David C. Ullrich
    Aug 27 at 14:52

















  • Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
    – David C. Ullrich
    Aug 27 at 14:52
















Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
– David C. Ullrich
Aug 27 at 14:52





Yes, the zero vector is in the span of any set of vectors. But that really has very little to do with what you're trying to prove. You need to show that $span(v_1,v_2)=span(v_1,v_2,0)$; that doesn't follow from knowing that $0$ is in both spans.
– David C. Ullrich
Aug 27 at 14:52











4 Answers
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6
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Yes!



Span operation is monotone!



That is if $A subset B$ then $textspan(A) subset textspan(B)$



So $textspan(v_1,v_2) subset textspan(v_1,v_2,0)$



The reverse inclusion is also true, since.....?



Remembert that $textspan(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span






share|cite|improve this answer




















  • thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24

















up vote
3
down vote













The span of $v_1,v_2,0$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $v_1,v_2$.



Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.






share|cite|improve this answer






















  • thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24

















up vote
2
down vote













Yes of course by definition it easy to check that the sets $v_1, v_2, 0$ and $v_1, v_2$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.






share|cite|improve this answer






















  • thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24










  • @lohboys You are welcome! Bye
    – gimusi
    Aug 27 at 6:25

















up vote
0
down vote














all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_n+1$ is in the span of $v_1,v_2...v_n$, then the span of $v_1,v_2...v_n$ and the span of $v_1,v_2...v_n,v_n+1$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_n+1$ to your list of vectors doesn't add anything new to the span, if you can get $v_n+1$ from the other vectors.



In the example you give, the span of $v_1,v_2,0$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $v_1,v_2$






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    4 Answers
    4






    active

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    4 Answers
    4






    active

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    active

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    up vote
    6
    down vote













    Yes!



    Span operation is monotone!



    That is if $A subset B$ then $textspan(A) subset textspan(B)$



    So $textspan(v_1,v_2) subset textspan(v_1,v_2,0)$



    The reverse inclusion is also true, since.....?



    Remembert that $textspan(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span






    share|cite|improve this answer




















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24














    up vote
    6
    down vote













    Yes!



    Span operation is monotone!



    That is if $A subset B$ then $textspan(A) subset textspan(B)$



    So $textspan(v_1,v_2) subset textspan(v_1,v_2,0)$



    The reverse inclusion is also true, since.....?



    Remembert that $textspan(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span






    share|cite|improve this answer




















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24












    up vote
    6
    down vote










    up vote
    6
    down vote









    Yes!



    Span operation is monotone!



    That is if $A subset B$ then $textspan(A) subset textspan(B)$



    So $textspan(v_1,v_2) subset textspan(v_1,v_2,0)$



    The reverse inclusion is also true, since.....?



    Remembert that $textspan(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span






    share|cite|improve this answer












    Yes!



    Span operation is monotone!



    That is if $A subset B$ then $textspan(A) subset textspan(B)$



    So $textspan(v_1,v_2) subset textspan(v_1,v_2,0)$



    The reverse inclusion is also true, since.....?



    Remembert that $textspan(A)$ means intersection of all subspaces containing $A$ and since this intersection is again a subspace, zero is in that space . So we dont worrying about including zero in span







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Aug 27 at 6:21









    LDM

    730314




    730314











    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24
















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24















    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24




    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24










    up vote
    3
    down vote













    The span of $v_1,v_2,0$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $v_1,v_2$.



    Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.






    share|cite|improve this answer






















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24














    up vote
    3
    down vote













    The span of $v_1,v_2,0$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $v_1,v_2$.



    Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.






    share|cite|improve this answer






















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24












    up vote
    3
    down vote










    up vote
    3
    down vote









    The span of $v_1,v_2,0$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $v_1,v_2$.



    Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.






    share|cite|improve this answer














    The span of $v_1,v_2,0$ is the set of all linear combinations of these three vectors, and this certainly equals the set of all linear combinations of $v_1,v_2$ which is the span of $v_1,v_2$.



    Perhaps you have confused this with another matter in your question: Indeed, the zero vector is in the span of any set of vectors, because by using zero coefficients for each vector we generate the zero vector. You need to distinguish between (1) the span of a set of vectors and (2) the vectors in the set used to span.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Aug 27 at 6:42

























    answered Aug 27 at 6:20









    Jasper Loy

    2806




    2806











    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24
















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24















    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24




    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24










    up vote
    2
    down vote













    Yes of course by definition it easy to check that the sets $v_1, v_2, 0$ and $v_1, v_2$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.






    share|cite|improve this answer






















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24










    • @lohboys You are welcome! Bye
      – gimusi
      Aug 27 at 6:25














    up vote
    2
    down vote













    Yes of course by definition it easy to check that the sets $v_1, v_2, 0$ and $v_1, v_2$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.






    share|cite|improve this answer






















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24










    • @lohboys You are welcome! Bye
      – gimusi
      Aug 27 at 6:25












    up vote
    2
    down vote










    up vote
    2
    down vote









    Yes of course by definition it easy to check that the sets $v_1, v_2, 0$ and $v_1, v_2$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.






    share|cite|improve this answer














    Yes of course by definition it easy to check that the sets $v_1, v_2, 0$ and $v_1, v_2$ span the same subspace and that the zero vector is always in the span of any non empty set of vectors.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Aug 27 at 6:26

























    answered Aug 27 at 6:18









    gimusi

    70.7k73786




    70.7k73786











    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24










    • @lohboys You are welcome! Bye
      – gimusi
      Aug 27 at 6:25
















    • thanks for the prompt reply!
      – lohboys
      Aug 27 at 6:24










    • @lohboys You are welcome! Bye
      – gimusi
      Aug 27 at 6:25















    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24




    thanks for the prompt reply!
    – lohboys
    Aug 27 at 6:24












    @lohboys You are welcome! Bye
    – gimusi
    Aug 27 at 6:25




    @lohboys You are welcome! Bye
    – gimusi
    Aug 27 at 6:25










    up vote
    0
    down vote














    all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




    It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_n+1$ is in the span of $v_1,v_2...v_n$, then the span of $v_1,v_2...v_n$ and the span of $v_1,v_2...v_n,v_n+1$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_n+1$ to your list of vectors doesn't add anything new to the span, if you can get $v_n+1$ from the other vectors.



    In the example you give, the span of $v_1,v_2,0$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $v_1,v_2$






    share|cite|improve this answer
























      up vote
      0
      down vote














      all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




      It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_n+1$ is in the span of $v_1,v_2...v_n$, then the span of $v_1,v_2...v_n$ and the span of $v_1,v_2...v_n,v_n+1$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_n+1$ to your list of vectors doesn't add anything new to the span, if you can get $v_n+1$ from the other vectors.



      In the example you give, the span of $v_1,v_2,0$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $v_1,v_2$






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote










        all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




        It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_n+1$ is in the span of $v_1,v_2...v_n$, then the span of $v_1,v_2...v_n$ and the span of $v_1,v_2...v_n,v_n+1$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_n+1$ to your list of vectors doesn't add anything new to the span, if you can get $v_n+1$ from the other vectors.



        In the example you give, the span of $v_1,v_2,0$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $v_1,v_2$






        share|cite|improve this answer













        all vectors in a span can be multiplied by 0 to get the zero vector, so shouldn't the zero vector be in every and any span of vectors?




        It's not entirely clear what your thinking is. What you might be getting at is that given any $v_1,v_2...v_n$, if $v_n+1$ is in the span of $v_1,v_2...v_n$, then the span of $v_1,v_2...v_n$ and the span of $v_1,v_2...v_n,v_n+1$ are the same. The span of a set of vectors is, by definition, closed over linear combinations; adding $v_n+1$ to your list of vectors doesn't add anything new to the span, if you can get $v_n+1$ from the other vectors.



        In the example you give, the span of $v_1,v_2,0$ is defined as the set of all vectors that can be written in the form $c_1v_1+c_2v_2+c_30$, for some scalars $c_1,c_2,c_3$. But if $v=c_1v_1+c_2v_2+c_30$, then clearly $v=c_1v_1+c_2v_2$, so $v$ is in the span of $v_1,v_2$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 27 at 15:16









        Acccumulation

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