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Homework Help: Subspace notation help

  1. Oct 24, 2004 #1
    Hi everyone, I was hoping someone could help me with something. Could someone explain to me exactly what this expression means:

    H = {(a+b) + (a - 2b)t + bt^2 | a E R, b E R}

    the purpose is to find the dimension of the given subspace, which I know how to do, I have just never seen this notation, so I'm not exactly sure what this expresssion is telling me about the subspace. since a and e are real numbers, does that mean t is not necessarily a real number? and is the first part of the expression a single vector? or is it some sort of conglomeration of multiple vectors? Thanks for your help
    Davy
     
  2. jcsd
  3. Oct 24, 2004 #2

    Galileo

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    I think you are looking at a vector space of polynomials (of degree 2 or higher).
    A set basis vectors could be: [itex]\{1,t,t^2,t^3,...,t^n\}[/itex]
    Then [tex]H=\{(a+b)+(a-2b)t +bt^2|a \in \mathbb{R}, b \in \mathbb{R}\}[/tex]
    would be a subspace of the polynomial space.
    So, formally speaking, [itex]t[/itex] is not a number, but a vector.
     
  4. Oct 24, 2004 #3

    HallsofIvy

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    You also posted this under "linear algebra" and were given very good answers there.

    The "notation" means that H is the subset of all quadratic polynomials
    α+ βt+ γt2 such that α= a-b, β= a- 2b, and γ= b.

    The set of all quadratic polynomials has dimension 3 specifically because we can select any of the coefficients α, β, γ arbitrarily: a basis is { 1, t, t2}.

    Here the coefficients depend upon only two numbers. Pick a or b arbitrarily and then you can calculate α, β, γ .

    In particular, if you take a= 1, b= 0, the polynomial is 1+ t and if you take a= 0, b= 1, the polynomial is 1- 2t+ t2. A basis for H is {1+t, 1- 2t+ t2} so H has dimension 2.
     
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