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Vector spaces and subspaces

  1. Jan 28, 2010 #1
    1. The problem statement, all variables and given/known data
    For each of the following subsets U of the vector space V decide whether or not U is a
    subspace of V . Give reasons for your answers. In each case when U is a subspace, find a
    basis for U and state dim U


    2. Relevant equations

    [tex]V=P_{3} ; U=\left\{p\in\ P_{3}:p(0)-p'(0)=0\right\}[/tex]

    3. The attempt at a solution

    so i set it out as (a_3p^3+...+a_1p^1+a_0)-(3a_3p^2+2a_2p+a_1)=0 then rearranged all that to get; a_3p^3+...+(a_1-2a_2)p^1+(a_0-a_1)=0

    Now i'm not sure what to do next. I think its because i'm not comfortable with the idea of polynomials as regards to vector spaces. does the fact that a couple of them share coefficients as it were mean that they would be dependent and so reduce the dimension? i guess making it a one dimensional subspace? All help and explanations much appreciated.

    p.s. another subset is [tex]x=(x_{1},x_{2},x_{3}): x^{2}_{3}=x^{2}_{1}+x^{2}_{2}[/tex]
    I want to say this is not a subspace because of the fact its not a linear equation? would that be correct? Doesn't feel like very rigorous reasoning so it feels wrong.
     
  2. jcsd
  3. Jan 28, 2010 #2

    radou

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    What is the condition (let's say, a "test" condition) for U to be a subspace of some vector space V?
     
  4. Jan 28, 2010 #3
    that the elements of U are closed under addition and scalar multiplication operations and it contains the zero vector.
     
  5. Jan 28, 2010 #4

    radou

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    Correct. Any idea of how to test it? Take two vectors (polynomials) from U, let's say p and q. Is their linear combination αp + βq also in U? Is the zero vector in U (i.e. does it satisfy the condition on the coefficients implied by p(0) - p'(0) = 0)?
     
  6. Jan 28, 2010 #5
    Ok right so the zero vector would be in U right? Just the one with all zero coefficients?

    And the condition for p(0) - p'(0) = 0 is that [tex]p_{0}=p_{1}[/tex] so provided you chose polynomials P and Q that satisfied this then (ap+bq)-(ap+bq)' = ap+bq-ap'-bq' = ap-ap' + bq-bq' = a(p-p')+b(q-q') = a0 + b0 so that works to.

    Am i on the right track so far?

    So then given the condition that [tex]p_{0}=p_{1}[/tex] does that reduce the dimension from 3 to 2?
     
  7. Jan 28, 2010 #6
    To conclude [tex]W \subset V[/tex] is a subspace of [tex]V[/tex], it suffices to verify that for arbitrary scalar [tex]\alpha[/tex] and arbitrary vectors [tex]x,y \in W[/tex], that [tex](\alpha\cdot x + y)\in W[/tex] as well.

    So, consider two vectors [tex]p, q \in U[/tex], and either (1) prove that for any scalar [tex]\alpha \in \mathbb{R}[/tex] that [tex](\alpha\cdot p + q)[/tex] is also in [tex]U[/tex], or (2) show a case where it need not be.
     
  8. Jan 28, 2010 #7

    vela

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    Yes, you've shown that the subset is closed under scalar multiplication and vector addition, so it's a subspace.

    From 4 to 3, actually. P3 is spanned by {1, x, x2, x3}, so it's a four-dimensional space.
     
  9. Jan 29, 2010 #8
    Yeah that makes more sense silly me. Thanks all!
     
  10. Jan 29, 2010 #9
    Another question! V is a vector space in R4. And U={x=[x1,x2,x3,x4]: x1-2x2-3x3+x4=0}

    so I'm happy this is a sunspace but I have to determine a basis and dimension.

    Am I correct thinking that clearly there are non zero scalars such that x1+...+x4 equals zero, namely 1,-2,-3,1. So they are linearly dependant? How do I make a basis out of that?
     
  11. Jan 29, 2010 #10

    Mark44

    Staff: Mentor

    Some corrections: V is a subspace of R4. R4 is a vector space in its own right. Actually there doesn't seem to be a need for V, since it isn't mentioned in the rest of the problem.

    U is a subspace (as it turns out) of R4 such that if x is in U, then x1 - 2x2 - 3x3 +x4 = 0, where x1, x2, x3, and x4 are the coordinates of vector x. They are not vectors, so it makes no sense to talk about them being linearly dependent/independent.

    This equation represents a "hyperplane" in R4. Have you seen any other problems where an equation such as the one above is the starting point for finding a basis for the subspace determined by that equation?
     
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