# Let $V$ be the inner product space consisting of linear polynomials

I am stuck with the following problem:

Let $V$ be the inner product space consisting of linear polynomial, $p\colon[0,1]\rightarrow\mathbb R$ (i.e. $V$ consists of polynomials $p$ of the form $p(x)=ax+b,a,b \in \mathbb R$) with the inner product defined by $\langle p,q\rangle=\int_{0}^{1}p(x)q(x)dx$ for $p,q \in V$. Then an orthonormal basis of $V$ is which of the following:
$1.\{1,x\}$
$2.\{1,x\sqrt 3\}$
$3.\{1,(2x-1)\sqrt 3\}$
$4.\{1,x-1/2\}$

Can someone point me in the right direction?Thanks in advance for your time.

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Do you know what an orthonormal basis is? – Stefan Feb 2 '13 at 17:20
+1 nice question. – Babak S. Feb 4 '13 at 17:25

Working just from the definitions, an orthonormal basis requires (1) orthogonality, (2) each member has unit norm, (3) the set forms a basis for the space.

To check (1) for the proposed sets $\{f,g\}$ above, check if $\langle f,g\rangle=0$.

To check (2), see if each member of the proposed set satisfies $\|f\|=1\iff \langle f,f\rangle^{1/2}=1$.

(3) should be really quick.

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Thanks a lot sir.I have got it.I see that option $(3)$ satisfies the three conditions.So it will be the required answer. – user52976 Feb 2 '13 at 17:58

When you are stuck in front of a problem, the first thing to do is to ask yourself if you know precisely and understand the definitions of all the notions involved.

Here, an orthonormal basis of inner product space $V$ is a basis $(e_1,\dots,e_n)$ of $V$, whose elements are orthonormal, i.e. $$\langle e_i,e_j\rangle=\delta_{ij}=\begin{cases}1&\text{ if }i=j\\0&\text{ if }i\not=j.\end{cases}$$ So you simply have to check these two points for the four sets given, this is calculating some simple integrals (but tell us if this is what gives you trouble).

Note that by symmetry, you will here have to check at most 3 equations (2 for the norms and 1 for the orthogonality.

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