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Question 6.7.3: Let V be P2, with the inner product from Example 2, where t0...

Let V be P _{2} \text {, with the inner product from Example 2, where } t_{0}=0, t_{1}=\frac{1}{2}, \text { and } t_{2}=1 . \text { Let } p(t)=12 t^{2} \text { and } q(t)=2 t-1 . \text { Compute }\langle p, q\rangle \text { and }\langle q, q\rangle .

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\langle p, q\rangle=p(0) q(0)+p\left(\frac{1}{2}\right) q\left(\frac{1}{2}\right)+p(1) q(1).

=(0)(-1)+(3)(0)+(12)(1)=12.

\langle q, q\rangle=[q(0)]^{2}+\left[q\left(\frac{1}{2}\right)\right]^{2}+[q(1)]^{2}.

=(-1)^{2}+(0)^{2}+(1)^{2}=2.

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