I want to check numerically if a certain vector relation like $$ \alpha_1v_1+...+\alpha_kv_k=c \ (1)$$ holds (where $v_i,c$ are vectors of $100$ or more components). For this, I use least squares approximation, and to see the error, I evaluate the norm of the difference $$ \|\alpha_1v_1+...+\alpha_kv_k-c\|.$$ All $v_i$ and $c$ are computed numerically, so they are subject to errors. The size of the vectors is $100$ components.
When can I say that the error is small enough, i.e. the relation $(1)$ holds numerically?
I ask the question because I obtain errors between $0.2$ and $1$, which seem rather large, but if we look component wise, the average error is small.