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Math Help - Dimension and basis of the Vector Space

  1. #1
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    Dimension and basis of the Vector Space

    Find the dimension of the vector space V and give a basis for V

    V = {p(x) in P2 : p(0) = 0}

    *P2 denotes polynomials of degree less than or equal to 2

    Sol:

    standard basis of P2={1, x, x^2}

    let p(x)=a+bx+cx^2
    then p(0)=a+0+0 ====> a=0,

    do we through away 1 here?

    hence
    I got Vdim=2 and basis {x, x^2}, but have a feeling that its not correct
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  2. #2
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    Let v \in V. Then v is a polynomial of degree less than or equal to 2 with v(0)=0. Expressed this way v = a+bx+cx^2 where a,b,c are in the field over which V is a vector space. As you said this implies a = 0, so v = bx+cx^2. Therefore v (an arbitrary element of V) can be written as a linear combination of elements in {x,x^2} therefore {x,x^2} is a spanning set. Show this set is also linearly independent (by assuming ax + bx^2 = 0 and concluding that a = b = 0) and you've got yourself a basis.

    Or show that {x, x^2} is the minimal spanning set; that is removing any element destroys the spanning property.
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  3. #3
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    Why do you think that is not correct?
    If P(x)=ax+bx^2 then P(0)=0. Correct?
    What about P(x)=c+ax+bx^2,\;c\not=0 can P(0)=0?
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