Let $\{u_1,u_2,\ldots,u_n\}$ be the orthogonal (i.e., before the normalization) basis obtained from linearly independent vectors $\{v_1,v_2,\ldots,v_n\}$ by the Gram-Schmidt process, starting from $u_1=v_1$ and $$u_j=v_j-\sum_{k=1}^{j-1}\langle v_j,u_k/\| u_k\|\rangle u_k/\| u_k\|, \quad\text{ for }2\leq j\leq n.$$
From the linear independence, the resulting vectors will be non-zero. But intuitively, if $\{v_1,v_2,\ldots,v_n\}$ are close to linear dependent, some $u_j$ should be near $0$.
I wonder if we can assess how close $\|u_j\|$ are to $0$ (lower bound of $\|u_j\|$) in terms of how linear independent ${v_1,\dots,v_n}$ are. I thought of getting a lower bound of $\|u_j\|$ in terms of the smallest eigenvalue of the Gram matrix of $(v_1|\cdots|v_n)$, but no luck.