need help for subspaces

**mathlovet** · October 16th 2008, 03:09 PM

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}

**NonCommAlg** · October 16th 2008, 04:54 PM

Originally Posted by mathlovet

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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