Question: The usefulness of orthogonal and orthonormal bases becomes evident when considering the calculations for changing coordinates to a new basis. When the basis is orthogonal, calculating the coordinates is much simpler.

1. Given a vector \(\mathbf{v}\) and an orthogonal basis \(\mathcal{B}=\left\{\mathbf{u}_{1}, \ldots, \mathbf{u}_{n}\right\}\), show that \(\mathbf{v}\) can be written as \(\mathbf{v}_{B}=\left[\begin{array}{c}c_{1} \\ \vdots \\ c_{n}\end{array}\right]\)
   where the coordinates are given by \(c_{i}=\frac{\left\langle\mathbf{v}, \mathbf{u}_{i}\right\rangle}{\left\|\mathbf{u}_{i}\right\|^{2}}\)
2. Find the coordinates of vector \(\mathbf{v}=\left[\begin{array}{c}3 \\ -10\end{array}\right]\) with respect to basis \(\mathcal{S}=\left\{\left[\begin{array}{c}\frac{3}{\sqrt{13}} \\ -\frac{2}{\sqrt{13}}\end{array}\right],\left[\begin{array}{c}\frac{2}{\sqrt{13}} \\ \frac{3}{\sqrt{13}}\end{array}\right]\right\}\)

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