(i) The vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ in $\mathbb{R}^{3}$ satisfy $\mathbf{a}_{1} \cdot \mathbf{a}_{2} \times \mathbf{a}_{3} \neq 0$. Are $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ necessarily linearly independent? Justify your answer by a proof or a counterexample.

(ii) The vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$ in $\mathbb{R}^{n}$ have the property that every subset comprising $(n-1)$ of the vectors is linearly independent. Are $\mathbf{a}_{1}, \mathbf{a}_{2}, \ldots, \mathbf{a}_{n}$ necessarily linearly independent? Justify your answer by a proof or a counterexample.

(iii) For each value of $t$ in the range $0 \leqslant t<1$, give a construction of a linearly independent set of vectors $\mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3}$ in $\mathbb{R}^{3}$ satisfying

$$\mathbf{a}_{i} \cdot \mathbf{a}_{j}=\delta_{i j}+t\left(1-\delta_{i j}\right),$$

where $\delta_{i j}$ is the Kronecker delta.