Does bases being disjoint imply subspaces are independent?

Question

Let $V$ be a vector space. Let $\{W_i\}_{i=1}^n$ be $n$ subspaces of $V$ and let $\{\text{B}_i\}_{i=1}^n$ be their bases respectively.

Suppose $\text{B}_i $ are disjoint. Does that imply independence of $W_i $?

The other way round is trivial, i.e, if $\text{B}_i$ are not disjoint, then there exists a common non zero element $k$ and consequently, $-k$ is also common. Thus $k+(-k) = 0$ and this implies dependence of $W_i$.

I don't know how to approach this one though.

Any help will be appreciated.
Thanks in advance.

Answer 1

No, consider the one-dimensional space $\mathbb{R}$ with bases $\left\{1\right\}$ and $\left\{2\right\}$ respectively.

Answer 2

No. Take $\mathbb R$ as your (real) vector space, take $B_1=\{1\}$, and take $B_2=\{-1\}$.