Exercise 11.2.4

Consider $\mathbb{R}^{3}$ as an inner product space with the usual dot product. For each of the following bases of $\mathbb{R}^{3}$, state whether it is orthonormal, orthogonal, or neither.

1. $\left\{\left[\begin{array}{r}1\\ 0\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]\right\}$.

2. $\left\{\left[\begin{array}{r}1\\ 0\\ 1\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 1\end{array}\right],\left[\begin{array}{r}0\\ 0\\ 1\end{array}\right]\right\}$.

3. $\left\{\left[\begin{array}{r}1\\ 0\\ 2\end{array}\right],\left[\begin{array}{r}0\\ 1\\ 0\end{array}\right],\left[\begin{array}{r}-2\\ 0\\ 1\end{array}\right]\right\}$.

4. $\left\{\left[\begin{array}{r}\frac{3}{5}\\ \frac{4}{5}\\ 0\end{array}\right],\left[\begin{array}{r}0\\ 0\\ -1\end{array}\right],\left[\begin{array}{r}\frac{4}{5}\\ -\frac{3}{5}\\ 0\end{array}\right]\right\}$.