Orthogonal projections and Fourier series

Consider $\mathbb{R}^{3}$ with the usual dot product. Let

Note that $\mathbf{u}_{1}$ and $\mathbf{u}_{2}$ are orthogonal. Find the best approximation of $\mathbf{v}$ in $\operatorname{span}\left\{\mathbf{u}_{1},\mathbf{u}_{2}\right\}$.

Consider $\mathbb{R}^{4}$ with the usual dot product. Let

Note that $\mathbf{u}_{1}$, $\mathbf{u}_{2}$, and $\mathbf{u}_{3}$ are orthogonal. Find the best approximation of $\mathbf{v}$ in $\operatorname{span}\left\{\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{3}\right\}$.

In the inner product space $V=C[-1,1]$, consider the function $f\in V$ given by

Find the closest approximation to $f$ by a polynomial of degree at most 0, 1, 2, 3, and 4. Graph both $f$ and the approximating polynomials.

In the inner product space $C[-\pi,\pi]$, consider the orthogonal set of functions from Example 11.26: $\left\{1,\sin x,\cos x,\sin 2x,\cos 2x,\sin 3x,\cos 3x,\ldots\right\}$. Let $f(x)=x^{2}$, where $x\in[-\pi,\pi]$. Find the Fourier series of $f$.