# Math Help - need help for subspaces

1. ## 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}

2. 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: $\dim_{\mathbb{R}}W=n-1. \ \ \ \Box$