?

Linear Algebra

Free Version

Upgrade subject to access all content

Difficult

Nearest Point to a Subsapce

LINALG-67DZ8D

Let $V$ be the linear span of the set of vectors $\{1, \sqrt{2}\cos 2\pi x, \sqrt{2}\sin 2\pi x\}$ where $0\leq x\leq 1$. Recall that $V$ is equipped with the inner product: $\langle u, v \rangle=\int_0^1u(x)v(x)\, dx$ whenever $u, v\in V$.

The nearest vector $f$ in $V$ to the vector $g(x)=e^x$ is

A

$0$

B

$e-1 +\cfrac{2(e-1)}{1+4\pi^2}\cos 2\pi x -\cfrac{4\pi(e-1)}{1+4\pi^2}\sin 2\pi x$

C

$e^x + \sin 2\pi x -2\cos 2\pi x$

D

$e^x\sin 2\pi x -e^{-x}\cos 2\pi x$