Consider the following vectors $v_1 = (1,2,-1)$, $v_2 = (3,-1,1)$, and $v_3 = (-3,8,-5)$
a) determine if they span $\mathbb{R}^3$?
I tried doing this question but when i put the vectors to a coefficient matrix the determinence I get is $0$. So i know its not a unique solution but it could no solution or infinite solution. My question is can you see if vectors span if the determinant was to be $0$?
and how would i know if the three vectors or linearly dependent or not?
Thanks in advance!
Best Answer
Since the determinant is $0$ (I didn't check it), the vectors $v_1$, $v_2$, and $v_3$ are linearly dependent. A set of $3$ linearly dependent vectors cannot span a $3$-dimensional vector space, such as $\mathbb{R}^3$.