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Linear Algebra

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Difficult

Even Orthogonal to Odd

LINALG-V4HGSY

Let $V=\{h:[-1,1]\rightarrow\mathbb{R}\ \bigr|\ h\text{ is continuous}\}$.

Then we may define an inner product on $V$ by $\displaystyle\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)\, dx$.

Which pair of vectors (functions) are orthogonal with respect to this inner product?

A

$ f(x)=e^{x^2}$, $g(x)=x^6$

B

$f(x)=e^{x^2}$, $g(x)=cos(\pi x)$

C

$f(x)=e^{x^2}$, $g(x)=sin(\pi x)$

D

$f(x)=x^5$, $g(x)=sin(\pi x)$