[Math] Equivalent basis of a subspace

Question

How to check whether two sets of vectors $\left\{ \begin{array}{cccc}
v_{1} & v_{2} & \cdots & v_{m}\end{array}\right\} $ and $\left\{ \begin{array}{cccc}
u_{1} & u_{2} & \cdots & u_{m}\end{array}\right\}$ where $v_{i},u_{i}\in R^{n}$
are the basis of the same subspace of dim m?

Answer 1

Check that the the $v_k$ are linearly independent and that each $v_k$ can be written in terms of the $u_i$. (Or the other way around, with $u,v$ interchanged.)

That is, check that if $\sum_k\alpha_kv_k = 0$ then $\alpha_k = 0$.

And check that for each $k$, there are constants $\beta_i$ such that $v_k = \sum_i \beta_i u_i $.