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Math Help - need help for subspaces

  1. #1
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    need help for subspaces

    let a (belong to) R be fixed. Determine the dimension of the subspace W of P(n) (R) defined by w={f(belongs to) P(n) R : f(a) = 0}
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  2. #2
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    Quote Originally Posted by mathlovet View Post
    let a (belong to) R be fixed. Determine the dimension of the subspace W of P(n) (R) defined by w={f(belongs to) P(n) R : f(a) = 0}
    i think by V=P_n(\mathbb{R}) you mean the set of all polynomials of degree at most n with real coefficients. define \varphi: V \longrightarrow \mathbb{R} by \varphi(f)=f(a). clearly \varphi is a surjective homomorphism and \ker \varphi = W.

    thus V/W \simeq \mathbb{R}. hence: n - \dim_{\mathbb{R}} W = \dim_{\mathbb{R}}V - \dim_{\mathbb{R}} W = \dim_{\mathbb{R}}\mathbb{R}=1, which gives us: \dim_{\mathbb{R}}W=n-1. \ \ \ \Box
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