# Linear span and linear combinations?

You have a set of vectors $v_1, v_2, v_3,\dots, v_m$. You also have the set of vectors $r_1, r_2, \dots, r_m$. Also the vectors in the second set are a linear combination of the vectors in the first set. For example, for any $1 \le k \le m$, $r_k = \sum_{i = 1} ^ {m}a_iv_i$ for some scalars $a_1, a_2, a_3 ,\dots, a_m$. How can I show that the linear span of $v_1, v_2, v_3,\dots, v_m$ is the same as the linear span of $r_1, r_2, \dots, r_m$?

This isn't homework, and I don't really have any work to show.

• Your statement is not true, for example if all $a_i=0$. Also it is better to use $a_{ij}$ instead, or it seems to me that all $r_k$ are the same vector
– user99914
Oct 27 '13 at 18:46

It's not true in general. Consider the case where the $v_i$'s are linearly independent, but the $r_i$'s aren't. (For example, the trivial case where $r_i = 0$ for all $i$.) Then the span of the $v_i$'s has dimension $m$ and the span of the $r_i$'s has smaller dimension.
You must check if the {$r_{i}$} have the same number of indepentent in between them as the $u_{i}$.then if so,u just multiply and you are over!