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`I'm having some trouble on this one: consider the set V of all polynomials of degree 2 or less, and let $$\langle u, v \rangle = \int_0^1 \! p(x) \!q(x) \, \mathrm{d} x$$

Find a matrix A such that $$\langle u, v \rangle = u^\top Av$$ and find an orthonormal basis of V with respect to this inner product.

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    $u$ is a polynomial, so what do you mean by $u^T$? And $v$ is a polynomial, so how can you multiply a matrix $A$ by a polynomial?2012-11-28
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    Also, your formula for $\langle u, v \rangle$ doesn't appear to use $u$ or $v$.2012-11-28
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    When I'm referring to $u^{\top}$, I'm referring to writing the polynomial as columns, so that for example $3x^2+6x+2$ would be written as $(3, 6, 2)^{\top}$. $\langle u, v \rangle$ refers to the inner product of a polynomial of degree 2 or less represented by u and another polynomial represented by v. In the integral, they are written as $p(x)$ for u and $q(x)$ for v.2012-11-28

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Let $\{e_0,e_1,e_2\}$ be the standard basis for $\mathbb R^3$. Note that $A$ must satisfy \begin{align*} \langle x^i, x^j \rangle &= e_i^T A e_j \\ &= A_{ij} \end{align*} for $0 \leq i,j \leq 2$. This tells us how to compute the entries of $A$.

You can use the Gram-Schmidt process to find an orthonormal basis of $V$ with respect to this inner product.