orthonormal set depending on variable

Question

Let $A(t) = \{x_1(t), x_2(t),..., x_n(t) \}$ with $0 \leq t \leq 1$ where $x_i(t) \in \mathbb{R}^n, \forall i$. I would like to construct a set $A(t)$ such that

1. $A(t) $ is an orthonormal set, i.e. $x_i(t)^T x_j(t) = \delta_{i,j}$, for all $t$ and for all $i,j \in \{1, 2, ...,n\}$
2. $\forall i: x_i(t)$ is a smooth function and $x_i$ is not constant
3. $\forall i:$ there is no subspace $V \subset \mathbb{R}^n$ with $dim(V) < r$ such that $x_i(t) \in V, \forall t$ for $r < n/10$

Are there easy ways to construct such sets? For $n = 2$ it can be done by $$A(t) = \{ \begin{bmatrix} \sin(t)\\ \cos(t) \end{bmatrix}, \begin{bmatrix} -\cos(t)\\ \sin(t) \end{bmatrix} \}$$ , but I would like to construct it for $n >> 2$

Answer 1

There are several ways to do this. One way that works for even values of $n$ would be to have $$x_1=\begin{bmatrix}\sin t\\\cos t \\0\\0\\0\\\vdots\\0\end{bmatrix}, x_2 = \begin{bmatrix}-\cos t\\\sin t \\0\\0\\0\\\vdots\\0\end{bmatrix}, x_3=\begin{bmatrix}0\\0\\\sin t\\\cos t \\0\\\vdots\\0\end{bmatrix}, x_4=\begin{bmatrix}0\\0\\-\cos t\\\sin t \\0\\\vdots\\0\end{bmatrix}, x_5=\begin{bmatrix}0\\0\\0\\0\\\sin t\\\cos t \\\vdots\\0\end{bmatrix}, x_6=\begin{bmatrix}0\\0\\0\\0\\-\cos t\\\sin t \\\vdots\\0\end{bmatrix}$$ and so on.