Results 1 to 6 of 6

Thread: Vector subspaces

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    8

    Vector subspaces

    I am having difficulty with vector subspaces. It would be helpful if someone could give me an answer to at least one of these two questions.

    1.

    Prove that $\displaystyle W_1= \{(a_1,a_2 \cdot \cdot \cdot ,a_n) \in F^n : a_1+a_2\ + \cdot \cdot \cdot a_n =0 \} $ is a subspace of F^n, but $\displaystyle W_2= \{(a_1,a_2\,\cdot \cdot \cdot a_n) \in F^n : a_1+a_2 + \cdot \cdot \cdot a_n =1 \} $ is not


    2.

    Is the set W = { f(x) $\displaystyle \in $ P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Mar 2008
    From
    Pennsylvania, USA
    Posts
    339
    Thanks
    46
    $\displaystyle W_1= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2+...+ a_n =0 \}$


    If $\displaystyle a=(a_1,a_2,...,a_n)\in W_1 $ , $\displaystyle b=(b_1,b_2,...,b_n) \in W_1 $ and $\displaystyle t,u \in \mathbb{F} $ then

    $\displaystyle
    (ta+ub)_1 + (ta+ub)_2 + ... + (ta+ub)_n = (ta_1+ub_1)+(ta_2+ub_2)+...+ (ta_n+ub_n) $

    $\displaystyle = t(a_1+a_2+...+a_n) + u(b_1+b_2+...+b_n) $

    $\displaystyle
    = t\cdot0+u\cdot 0
    $

    $\displaystyle
    = 0
    $


    Therefore, $\displaystyle ta+ub \in W_1 \implies W1 $ is a subspace.
    Last edited by abender; Jan 11th 2010 at 12:04 PM. Reason: forgot a comma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Mar 2008
    From
    Pennsylvania, USA
    Posts
    339
    Thanks
    46
    $\displaystyle
    W_2= \{(a_1,a_2,...,a_n) \in F^n : a_1+a_2 +...+a_n =1 \}
    $


    $\displaystyle W_2 $ is not a vector space because $\displaystyle (0,0,...,0) \not\in W_2$. Note that $\displaystyle (0,0,...,0) $ is simply the zero vector.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,769
    Thanks
    3027
    Quote Originally Posted by Matharch View Post
    2.[/COLOR]

    Is the set W = { f(x) $\displaystyle \in $ P(F): f(x)=0 or f(x) has a degree n} a subspace of P(F) if n ≥ 1? Justify your answer[/COLOR]
    [/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. $\displaystyle g(x)= -x^2+ 3$ is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2010
    Posts
    8
    Thanks guys! It's so simple, I'm almost ashamed that I posted here.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Jun 2009
    From
    Africa
    Posts
    641

    Smile

    Quote Originally Posted by HallsofIvy View Post
    [/tex]f(x)= x^2+ 1[/tex] is a polynomial of degree 2. $\displaystyle g(x)= -x^2+ 3$ is a polynomial of degree 2 also. Is there sum a polynomial of degree 2? Is the set of "polynomials of degree 2" closed under addition?
    i didn't think that the second question could be answered this way.
    the "Or" made me confused,since it means union.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Vector Subspaces
    Posted in the Differential Geometry Forum
    Replies: 20
    Last Post: Oct 15th 2011, 02:45 PM
  2. Vector Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 13th 2010, 08:00 AM
  3. Intersection of vector subspaces
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Mar 30th 2009, 08:04 AM
  4. Vector Subspaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 26th 2008, 11:31 AM
  5. Vector Subspaces
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 10th 2007, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum