Results 1 to 4 of 4

Thread: [SOLVED] Vector space

  1. #1
    Neoxl
    Guest

    [SOLVED] Vector space

    I am sure this is easy but I have been trying all day and just don't understand the question.

    Determine whether the set, together with the indicated operations, is a vector space.

    1. The set {(x, -x) : x is a real number} with the standard operations.

    2. The set {(x, y) : x, y are a real numbers such that x > y} with the standard operations.

    Any help would be appreciated. Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Neoxl View Post

    1. The set {(x, -x) : x is a real number} with the standard operations.
    Let $\displaystyle S = \{(x,-x)\}$. Now if $\displaystyle a\in S, \ b\in S$ then $\displaystyle a=(x,-x)$ and $\displaystyle b=(y,-y)$ so $\displaystyle a+b = (x+y,-(x+y))$ so $\displaystyle a+b$ in $\displaystyle S$. Similarly $\displaystyle ka$ in $\displaystyle S$ where $\displaystyle k\in \mathbb{R}$. So it is a vector space.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    5
    Quote Originally Posted by ThePerfectHacker View Post
    Let $\displaystyle S = \{(x,-x)\}$. Now if $\displaystyle a\in S, \ b\in S$ then $\displaystyle a=(x,-x)$ and $\displaystyle b=(y,-y)$ so $\displaystyle a+b = (x+y,-(x+y))$ so $\displaystyle a+b$ in $\displaystyle S$. Similarly $\displaystyle ka$ in $\displaystyle S$ where $\displaystyle k\in \mathbb{R}$. So it is a vector space.
    don't you have to verify a list of like 10 axioms or something. that's what we had to do in my linear algebra class. i guess if the standard operations hold, then verifying that it is closed under addition and scalar multiplication is fine, but i don't know
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by Jhevon View Post
    don't you have to verify a list of like 10 axioms or something. that's what we had to do in my linear algebra class. i guess if the standard operations hold, then verifying that it is closed under addition and scalar multiplication is fine, but i don't know
    You do not need to. For example, since $\displaystyle \mathbb{R}^2$ is commutative then certainly any subset! So it just comes down to checking the closure of addition and scalar multiplication.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Banach space with infinite vector space basis?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Mar 24th 2011, 06:23 PM
  2. Replies: 15
    Last Post: Jul 23rd 2010, 11:46 AM
  3. [SOLVED] Subspace of a vector space
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: May 2nd 2010, 08:07 AM
  4. Replies: 3
    Last Post: Jun 29th 2009, 11:57 AM
  5. [SOLVED] Help in Vector space: Linearly independent!
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Feb 19th 2009, 10:52 PM

/mathhelpforum @mathhelpforum