2
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ 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 $\endgroup$
    – user99914
    Oct 27 '13 at 18:46
1
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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!

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.