Let V be a vector $\mathbb{R}$-space of all continuous $f(x)$ of one real variable defined on $[0,2]$. And let W denote the set of all $f(x) \in V$ s.t. the degree of $f(x) \lt 3, \: f(1)=0$.
Show that W is a vector subspace of V and find a basis and the $dim(W)$. We need to satisfy the conditions $\alpha f(x) \in W, \: \forall f(x) \in W$, which is straightforward since scalars will not change the degree of the polynomial and $\alpha f(1) = \alpha 0 = 0$, and that $f(x)+g(x) \in W, \: \forall f(x),g(x) \in W$. Let $f(x) = \alpha x^2 + \beta x + \gamma$ and $g(x) = \mu x^2 + \lambda x + \delta$, then $f(x) + g(x) = \alpha x^2 + \beta x + \gamma + \mu x^2 + \lambda x + \delta = (\alpha + \mu)x^2 + (\beta + \lambda)x + (\gamma + \delta) \in W$. Hence W is a vector subspace of V.
How do I go about finding the basis of W?
Also if I define V as an inner product space with $\big<f(x),g(x)\big>=\int_0^2 f(x)g(x)dx$, how do I find an orthonormal basis of W and the projection of $h(x)=2x^3-4x+1$ onto subspace W?
Best Answer
I assume that $W$ is meant to be the space of polynomials of degree at most 2, and with a root at 1. Is that right?
To find a basis, consider the fundamental theorem of algebra. The elements of $W$ must all have $(x-1)$ as a factor. Consider the monomial $u=(x-1)$, and construct a basis from there that is analogous to the usual basis (which would be an excellent basis for constructing the space of polynomials with a root at zero).
For your second question, once you have found a basis, you just use the Gramm-Schmidt process. Just remember that the projection of $\mathbf{v}$ onto the one dimensional vector space spanned by the unit vector $\mathbf{u}$ is just $$\langle \mathbf{u},\mathbf{v}\rangle \mathbf{u}$$ The vector $$\mathbf{v}'=\mathbf{v}-\langle \mathbf{u},\mathbf{v}\rangle \mathbf{u}$$ (often called the error term) is therefore orthogonal to $\mathbf{u}$, so $\{\mathbf{v}',\mathbf{u}\}$ is an orthogonal basis for the space $\textrm{Span}(\mathbf{u},\mathbf{v})$.
To project $\mathbf{w}$ onto the two dimensional space $\textrm{Span}(\mathbf{u},\mathbf{v})$, just take the sum of the projections of $\mathbf{w}$ onto the orthogonal basis vectors $\mathbf{u}$ and $\mathbf{v}$.