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Let $V_1, V_2, dots, V_n$ be a collection of vector subspaces in $mathbb R^n$. For each $j=1, dots, n$, $dim(V_j) = m$ with $2 le m < n$. We also have the condition: for any collection of $lceil{frac n m}rceil$ vector spaces from ${V_1, dots, V_n}$, then $V_{k_1} + dots + V_{k_{lceil frac n m rceil}} = mathbb R^n$. Suppose we construct a basis $U = {u_1, dots, u_n}$ of $mathbb R^n$ in the manner: $u_j in V_j$ for each $j$. Now suppose we construct another basis $W = {w_1, dots, w_n}$ in the same manner, i.e., $w_j in V_j$ for each $j$. I am wondering whether $U$ is connected with $W$ in the sense: there is a path $gamma = gamma_1 times gamma_2 times dots times gamma_n$, where each $gamma_j: [0,1] to V_j$ is a continuous path connecting $v_j$ and $w_j$ in $V_j$ and for each $t$: $gamma(t)$ forms a basis for $mathbb R^n$. We assume the basis ${v_j}$ and ${w_j}$ have the same orientation.

The basis can be identified by $GL_n(mathbb R)_+$ or $GL_n(mathbb R)_-$ and we know they are connected. But is there a way to guarantee on the path, each column vector only varies in the corresponding subspace?

I asked a similar question here Constructing a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces
. The conditions are stronger here.

An example of $V_1, dots, V_n$: suppose $n=5$, $m=2$. The construction I have in mind is: $V_i = text{span} ( (1, a_i, 0, 0, 0), (0, 0, 1, a_i, a_i^2))$. As long as $a_1 neq dots neq a_5 neq 0$, any three subspace would span $mathbb R^5$. For other cases, we can use similar idea.