Is a vector space a subspace of itself?vector space and its subspaceProve: The set of all polynomials p with p(2) = p(3) is a vector spaceProving a vector space over itself have no subspacesVector Space vs SubspaceProving a subset is a subspace of a Vector SpaceAre all vector spaces also a subspace?Understand the definition of a vector subspaceProve that, with vector addition and scalar multiplication well-defined, $V/W$ becomes a vector space over $k$.Is the set of all exponential functions a subspace of the vector space of all continuous functions?I've seen two definitions of subspace; one involving vector spaces and one requiring linear combinations

Prime joint compound before latex paint?

Add an angle to a sphere

Is a vector space a subspace of itself?

Doomsday-clock for my fantasy planet

Landlord wants to switch my lease to a "Land contract" to "get back at the city"

What are the advantages and disadvantages of running one shots compared to campaigns?

extract characters between two commas?

How to move the player while also allowing forces to affect it

What is the command to reset a PC without deleting any files

Is Social Media Science Fiction?

Why do UK politicians seemingly ignore opinion polls on Brexit?

What is it called when one voice type sings a 'solo'?

Re-submission of rejected manuscript without informing co-authors

Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?

How could a lack of term limits lead to a "dictatorship?"

How can I add custom success page

What is the meaning of "of trouble" in the following sentence?

New order #4: World

Is it legal to have the "// (c) 2019 John Smith" header in all files when there are hundreds of contributors?

When blogging recipes, how can I support both readers who want the narrative/journey and ones who want the printer-friendly recipe?

How to deal with fear of taking dependencies

Lied on resume at previous job

Why doesn't a const reference extend the life of a temporary object passed via a function?

What is GPS' 19 year rollover and does it present a cybersecurity issue?



Is a vector space a subspace of itself?


vector space and its subspaceProve: The set of all polynomials p with p(2) = p(3) is a vector spaceProving a vector space over itself have no subspacesVector Space vs SubspaceProving a subset is a subspace of a Vector SpaceAre all vector spaces also a subspace?Understand the definition of a vector subspaceProve that, with vector addition and scalar multiplication well-defined, $V/W$ becomes a vector space over $k$.Is the set of all exponential functions a subspace of the vector space of all continuous functions?I've seen two definitions of subspace; one involving vector spaces and one requiring linear combinations













1












$begingroup$


We know that a subspace is a vector space that follows the same addition and multiplication rules as $Bbb V$, but is a vector space a subspace of itself?
Also, I'm getting confused doing the practice questions, on when we prove that something is a vector space by using the subspace test and when we prove V1 - V10.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    How do you define a subspace of a vector space?
    $endgroup$
    – Brian
    3 hours ago






  • 1




    $begingroup$
    Is a set a subset of itself?? What’s V1-V10?
    $endgroup$
    – J. W. Tanner
    2 hours ago







  • 1




    $begingroup$
    The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
    $endgroup$
    – Theo Bendit
    2 hours ago










  • $begingroup$
    As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
    $endgroup$
    – Xander Henderson
    1 hour ago















1












$begingroup$


We know that a subspace is a vector space that follows the same addition and multiplication rules as $Bbb V$, but is a vector space a subspace of itself?
Also, I'm getting confused doing the practice questions, on when we prove that something is a vector space by using the subspace test and when we prove V1 - V10.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    How do you define a subspace of a vector space?
    $endgroup$
    – Brian
    3 hours ago






  • 1




    $begingroup$
    Is a set a subset of itself?? What’s V1-V10?
    $endgroup$
    – J. W. Tanner
    2 hours ago







  • 1




    $begingroup$
    The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
    $endgroup$
    – Theo Bendit
    2 hours ago










  • $begingroup$
    As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
    $endgroup$
    – Xander Henderson
    1 hour ago













1












1








1


1



$begingroup$


We know that a subspace is a vector space that follows the same addition and multiplication rules as $Bbb V$, but is a vector space a subspace of itself?
Also, I'm getting confused doing the practice questions, on when we prove that something is a vector space by using the subspace test and when we prove V1 - V10.










share|cite|improve this question











$endgroup$




We know that a subspace is a vector space that follows the same addition and multiplication rules as $Bbb V$, but is a vector space a subspace of itself?
Also, I'm getting confused doing the practice questions, on when we prove that something is a vector space by using the subspace test and when we prove V1 - V10.







linear-algebra vector-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 16 mins ago









Eric Wofsey

192k14220352




192k14220352










asked 3 hours ago









mingming

4456




4456







  • 1




    $begingroup$
    How do you define a subspace of a vector space?
    $endgroup$
    – Brian
    3 hours ago






  • 1




    $begingroup$
    Is a set a subset of itself?? What’s V1-V10?
    $endgroup$
    – J. W. Tanner
    2 hours ago







  • 1




    $begingroup$
    The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
    $endgroup$
    – Theo Bendit
    2 hours ago










  • $begingroup$
    As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
    $endgroup$
    – Xander Henderson
    1 hour ago












  • 1




    $begingroup$
    How do you define a subspace of a vector space?
    $endgroup$
    – Brian
    3 hours ago






  • 1




    $begingroup$
    Is a set a subset of itself?? What’s V1-V10?
    $endgroup$
    – J. W. Tanner
    2 hours ago







  • 1




    $begingroup$
    The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
    $endgroup$
    – Theo Bendit
    2 hours ago










  • $begingroup$
    As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
    $endgroup$
    – Xander Henderson
    1 hour ago







1




1




$begingroup$
How do you define a subspace of a vector space?
$endgroup$
– Brian
3 hours ago




$begingroup$
How do you define a subspace of a vector space?
$endgroup$
– Brian
3 hours ago




1




1




$begingroup$
Is a set a subset of itself?? What’s V1-V10?
$endgroup$
– J. W. Tanner
2 hours ago





$begingroup$
Is a set a subset of itself?? What’s V1-V10?
$endgroup$
– J. W. Tanner
2 hours ago





1




1




$begingroup$
The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
$endgroup$
– Theo Bendit
2 hours ago




$begingroup$
The term "proper" subspace is often used to denote a subspace space that is not the entire vector space.
$endgroup$
– Theo Bendit
2 hours ago












$begingroup$
As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
$endgroup$
– Xander Henderson
1 hour ago




$begingroup$
As other commenters have noted, your question lacks context. Please edit your question to include more context, lest your question be closed. Please give a definition of a subspace. Please explain what V1 - V10 means. If you are working from a particular text, a citation to that text would be helpful, too.
$endgroup$
– Xander Henderson
1 hour ago










2 Answers
2






active

oldest

votes


















7












$begingroup$

Yes, every vector space is a vector subspace of itself, since it is a non-empty subset of itself which is closed with respect to addition and with respect to product by scalars.






share|cite|improve this answer









$endgroup$




















    2












    $begingroup$

    I'm guessing that V1 - V10 are the axioms for proving vector spaces.



    To prove something is a vector space, independent of any other vector spaces you know of, you are required to prove all of the axioms in the definition. Not all operations that call themselves $+$ are worthy addition operations; just because you denote it $+$ does not mean it is (for example) associative, or has an additive identity.



    There is a lot to prove, because there's a lot to gain. Vector spaces have a simply enormous amount of structure, and that structure gives us a really rich theory and powerful tools. If you have an object that you wish to understand better, and you can show it is a vector space (or at least, related to a vector space), then you'll instantly have some serious mathematical firepower at your fingertips.



    Subspaces give us a shortcut to proving a vector space. If you have a subset of a known vector space, then you can prove just $3$ properties, rather than $10$. We can skip a lot of the steps because somebody has already done them previously when showing the larger vector space is indeed a vector space. You don't need to show, for example, $v + w = w + v$ for all $v, w$ in your subset, because we already know this is true for all vectors in the larger vector space.



    I'm writing this, not as a direct answer to your question (which Jose Carlos Santos has answered already), but because confusion like this often stems from some sloppiness on the above point. I've seen many students (and, lamentably, several instructors) fail to grasp that showing the subspace conditions on a set that is not clearly a subset of a known vector space does not prove a vector space. The shortcut works because somebody has already established most of the axioms beforehand, but if this is not true, then the argument is a fallacy.



    You can absolutely apply the subspace conditions on the whole of a vector space provided you've proven it's a vector space already with axioms V1 - V10.






    share|cite|improve this answer









    $endgroup$













      Your Answer





      StackExchange.ifUsing("editor", function ()
      return StackExchange.using("mathjaxEditing", function ()
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      );
      );
      , "mathjax-editing");

      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


















      StackExchange.ready(
      function ()
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3180447%2fis-a-vector-space-a-subspace-of-itself%23new-answer', 'question_page');

      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      7












      $begingroup$

      Yes, every vector space is a vector subspace of itself, since it is a non-empty subset of itself which is closed with respect to addition and with respect to product by scalars.






      share|cite|improve this answer









      $endgroup$

















        7












        $begingroup$

        Yes, every vector space is a vector subspace of itself, since it is a non-empty subset of itself which is closed with respect to addition and with respect to product by scalars.






        share|cite|improve this answer









        $endgroup$















          7












          7








          7





          $begingroup$

          Yes, every vector space is a vector subspace of itself, since it is a non-empty subset of itself which is closed with respect to addition and with respect to product by scalars.






          share|cite|improve this answer









          $endgroup$



          Yes, every vector space is a vector subspace of itself, since it is a non-empty subset of itself which is closed with respect to addition and with respect to product by scalars.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          José Carlos SantosJosé Carlos Santos

          173k23133241




          173k23133241





















              2












              $begingroup$

              I'm guessing that V1 - V10 are the axioms for proving vector spaces.



              To prove something is a vector space, independent of any other vector spaces you know of, you are required to prove all of the axioms in the definition. Not all operations that call themselves $+$ are worthy addition operations; just because you denote it $+$ does not mean it is (for example) associative, or has an additive identity.



              There is a lot to prove, because there's a lot to gain. Vector spaces have a simply enormous amount of structure, and that structure gives us a really rich theory and powerful tools. If you have an object that you wish to understand better, and you can show it is a vector space (or at least, related to a vector space), then you'll instantly have some serious mathematical firepower at your fingertips.



              Subspaces give us a shortcut to proving a vector space. If you have a subset of a known vector space, then you can prove just $3$ properties, rather than $10$. We can skip a lot of the steps because somebody has already done them previously when showing the larger vector space is indeed a vector space. You don't need to show, for example, $v + w = w + v$ for all $v, w$ in your subset, because we already know this is true for all vectors in the larger vector space.



              I'm writing this, not as a direct answer to your question (which Jose Carlos Santos has answered already), but because confusion like this often stems from some sloppiness on the above point. I've seen many students (and, lamentably, several instructors) fail to grasp that showing the subspace conditions on a set that is not clearly a subset of a known vector space does not prove a vector space. The shortcut works because somebody has already established most of the axioms beforehand, but if this is not true, then the argument is a fallacy.



              You can absolutely apply the subspace conditions on the whole of a vector space provided you've proven it's a vector space already with axioms V1 - V10.






              share|cite|improve this answer









              $endgroup$

















                2












                $begingroup$

                I'm guessing that V1 - V10 are the axioms for proving vector spaces.



                To prove something is a vector space, independent of any other vector spaces you know of, you are required to prove all of the axioms in the definition. Not all operations that call themselves $+$ are worthy addition operations; just because you denote it $+$ does not mean it is (for example) associative, or has an additive identity.



                There is a lot to prove, because there's a lot to gain. Vector spaces have a simply enormous amount of structure, and that structure gives us a really rich theory and powerful tools. If you have an object that you wish to understand better, and you can show it is a vector space (or at least, related to a vector space), then you'll instantly have some serious mathematical firepower at your fingertips.



                Subspaces give us a shortcut to proving a vector space. If you have a subset of a known vector space, then you can prove just $3$ properties, rather than $10$. We can skip a lot of the steps because somebody has already done them previously when showing the larger vector space is indeed a vector space. You don't need to show, for example, $v + w = w + v$ for all $v, w$ in your subset, because we already know this is true for all vectors in the larger vector space.



                I'm writing this, not as a direct answer to your question (which Jose Carlos Santos has answered already), but because confusion like this often stems from some sloppiness on the above point. I've seen many students (and, lamentably, several instructors) fail to grasp that showing the subspace conditions on a set that is not clearly a subset of a known vector space does not prove a vector space. The shortcut works because somebody has already established most of the axioms beforehand, but if this is not true, then the argument is a fallacy.



                You can absolutely apply the subspace conditions on the whole of a vector space provided you've proven it's a vector space already with axioms V1 - V10.






                share|cite|improve this answer









                $endgroup$















                  2












                  2








                  2





                  $begingroup$

                  I'm guessing that V1 - V10 are the axioms for proving vector spaces.



                  To prove something is a vector space, independent of any other vector spaces you know of, you are required to prove all of the axioms in the definition. Not all operations that call themselves $+$ are worthy addition operations; just because you denote it $+$ does not mean it is (for example) associative, or has an additive identity.



                  There is a lot to prove, because there's a lot to gain. Vector spaces have a simply enormous amount of structure, and that structure gives us a really rich theory and powerful tools. If you have an object that you wish to understand better, and you can show it is a vector space (or at least, related to a vector space), then you'll instantly have some serious mathematical firepower at your fingertips.



                  Subspaces give us a shortcut to proving a vector space. If you have a subset of a known vector space, then you can prove just $3$ properties, rather than $10$. We can skip a lot of the steps because somebody has already done them previously when showing the larger vector space is indeed a vector space. You don't need to show, for example, $v + w = w + v$ for all $v, w$ in your subset, because we already know this is true for all vectors in the larger vector space.



                  I'm writing this, not as a direct answer to your question (which Jose Carlos Santos has answered already), but because confusion like this often stems from some sloppiness on the above point. I've seen many students (and, lamentably, several instructors) fail to grasp that showing the subspace conditions on a set that is not clearly a subset of a known vector space does not prove a vector space. The shortcut works because somebody has already established most of the axioms beforehand, but if this is not true, then the argument is a fallacy.



                  You can absolutely apply the subspace conditions on the whole of a vector space provided you've proven it's a vector space already with axioms V1 - V10.






                  share|cite|improve this answer









                  $endgroup$



                  I'm guessing that V1 - V10 are the axioms for proving vector spaces.



                  To prove something is a vector space, independent of any other vector spaces you know of, you are required to prove all of the axioms in the definition. Not all operations that call themselves $+$ are worthy addition operations; just because you denote it $+$ does not mean it is (for example) associative, or has an additive identity.



                  There is a lot to prove, because there's a lot to gain. Vector spaces have a simply enormous amount of structure, and that structure gives us a really rich theory and powerful tools. If you have an object that you wish to understand better, and you can show it is a vector space (or at least, related to a vector space), then you'll instantly have some serious mathematical firepower at your fingertips.



                  Subspaces give us a shortcut to proving a vector space. If you have a subset of a known vector space, then you can prove just $3$ properties, rather than $10$. We can skip a lot of the steps because somebody has already done them previously when showing the larger vector space is indeed a vector space. You don't need to show, for example, $v + w = w + v$ for all $v, w$ in your subset, because we already know this is true for all vectors in the larger vector space.



                  I'm writing this, not as a direct answer to your question (which Jose Carlos Santos has answered already), but because confusion like this often stems from some sloppiness on the above point. I've seen many students (and, lamentably, several instructors) fail to grasp that showing the subspace conditions on a set that is not clearly a subset of a known vector space does not prove a vector space. The shortcut works because somebody has already established most of the axioms beforehand, but if this is not true, then the argument is a fallacy.



                  You can absolutely apply the subspace conditions on the whole of a vector space provided you've proven it's a vector space already with axioms V1 - V10.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  Theo BenditTheo Bendit

                  20.7k12354




                  20.7k12354



























                      draft saved

                      draft discarded
















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid


                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.

                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function ()
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3180447%2fis-a-vector-space-a-subspace-of-itself%23new-answer', 'question_page');

                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      How should I use the fbox command correctly to avoid producing a Bad Box message?How to put a long piece of text in a box?How to specify height and width of fboxIs there an arrayrulecolor-like command to change the rule color of fbox?What is the command to highlight bad boxes in pdf?Why does fbox sometimes place the box *over* the graphic image?how to put the text in the boxHow to create command for a box where text inside the box can automatically adjust?how can I make an fbox like command with certain color, shape and width of border?how to use fbox in align modeFbox increase the spacing between the box and it content (inner margin)how to change the box height of an equationWhat is the use of the hbox in a newcommand command?

                      Doxepinum Nexus interni Notae | Tabula navigationis3158DB01142WHOa682390"Structural Analysis of the Histamine H1 Receptor""Transdermal and Topical Drug Administration in the Treatment of Pain""Antidepressants as antipruritic agents: A review"

                      Haugesund Nexus externi | Tabula navigationisHaugesund pagina interretialisAmplifica