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Math Help - Answer check (Vector space)

  1. #1
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    Answer check (Vector space)

    Hi, can someone just check my answer, please?

    Question:
    Two subspaces S and T of P_2(\mathbb{R}) are given by

    S = \lbrace f(x) ~| ~3f(0)+f'(0) = 0 \rbrace
    T = \lbrace f(x) ~| ~f(1) = 0 \rbrace.

    Obtain a non-trivial quadratic g = ax^2 + bx +c such that g \in S\cap T.

    My reponse:
    Let  g = ax^2 + bx +c \in P_2(\mathbb{R}),
    Then  g' = 2ax + b.

    For membership in S: 3c + b=0 ~ (1).
    For membership in T: a+b+c=0 ~ (2).

    From (1), b=-3c.
    From (2), a=-b-c.

    Let c = 1.
    \therefore b=-3 and a=3-1=2.

    So g=2x^2-3x+1 \in S\cap T is one such non-trivial quadratic.
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  2. #2
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    Quote Originally Posted by scorpion007 View Post
    Hi, can someone just check my answer, please?

    Question:
    Two subspaces S and T of P_2(\mathbb{R}) are given by

    S = \lbrace f(x) ~| ~3f(0)+f'(0) = 0 \rbrace
    T = \lbrace f(x) ~| ~f(1) = 0 \rbrace.

    Obtain a non-trivial quadratic g = ax^2 + bx +c such that g \in S\cap T.

    My reponse:
    Let  g = ax^2 + bx +c \in P_2(\mathbb{R}),
    Then  g' = 2ax + b.

    For membership in S: 3c + b=0 ~ (1).
    For membership in T: a+b+c=0 ~ (2).

    From (1), b=-3c.
    From (2), a=-b-c.

    Let c = 1.
    \therefore b=-3 and a=3-1=2.

    So g=2x^2-3x+1 \in S\cap T is one such non-trivial quadratic.
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