Results 1 to 3 of 3

Thread: [SOLVED] Prove polynomials are a generating system (vector spaces)

  1. #1
    Member
    Joined
    Mar 2009
    Posts
    90

    [SOLVED] Prove polynomials are a generating system (vector spaces)

    Let V be the vector space of the polynomials of degree $\displaystyle \leq $ 2 over $\displaystyle \mathbb{R}$.

    Prove that the polynomials:

    $\displaystyle p_1 = 2T^2 + T + 1 $
    $\displaystyle p_2 = 4T^2 + T$
    $\displaystyle p_3 = -2T^2 + 2T + 1$

    span, or are a generating system of, V.

    If I solve the system of linear equations- how do I prove that they span V?
    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 bmp05 View Post
    Let V be the vector space of the polynomials of degree $\displaystyle \leq $ 2 over $\displaystyle \mathbb{R}$.

    Prove that the polynomials:

    $\displaystyle p_1 = 2T^2 + T + 1 $
    $\displaystyle p_2 = 4T^2 + T$
    $\displaystyle p_3 = -2T^2 + 2T + 1$
    .
    You just need to show these polynomial are linearly independent.
    Say that $\displaystyle a_1p_1 + a_2p_2+a_3p_3 = \bold{0}$ where $\displaystyle \bold{0}$ is the zero-polynomial.
    This means that,
    $\displaystyle 2a_1+4a_2-2a_3=0$
    $\displaystyle a_1 + \ \ \ a_2 + 2a_3 = 0$
    $\displaystyle a_1 + \ \ \ \ \ \ \ \ \ + a_3 = 0$

    Argue that $\displaystyle a_1=a_2=a_3=0$ is the only solution.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Twig's Avatar
    Joined
    Mar 2008
    From
    Gothenburg
    Posts
    396
    Write the polynomials as vectors in $\displaystyle \mathbb{R}^{3} $ .

    $\displaystyle A= \left[\begin{matrix} 1 & 0 & 1\\ 1 & 1 & 2 \\ 2 & 4 & -2 \end{matrix}\right] $

    Row operations will show that this matrix A has 3 pivot positions. So it therefore spans $\displaystyle \mathbb{R}^{3} $ and by the isomorphism between $\displaystyle \mathbb{R}^{3} \mbox{ and } \mathbb{P}_{3} $ also $\displaystyle \mathbb{P}_{3} $ .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Vector spaces over division rings.
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: May 31st 2010, 04:22 PM
  2. Vector Spaces, Subspaces and Polynomials.
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Mar 3rd 2010, 09:30 AM
  3. Three quick questions - Vector spaces and polynomials
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 27th 2010, 06:08 AM
  4. Replies: 2
    Last Post: Feb 1st 2010, 01:08 PM
  5. [SOLVED] Vector Spaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 4th 2008, 03:55 AM

Search Tags


/mathhelpforum @mathhelpforum