# Which rows of vectors are linearly dependent or linearly independent?

Let $$v_1=(1,2,1,-1)$$, $$v_2=(-2,3,8,-3)$$, $$v_3=(-8,5,22,2)$$, $$v_4=(1,2,3,0)$$, $$v_5=(2,3,0,-1)$$ be row vectors in $$\mathbb R^4$$.

1. Are row vectors $$v_1,v_2,v_3$$ linearly independent?
2. Are row vectors $$v_1,v_2,v_3,v_4,v_5$$ linearly independent?

I understand that when vectors are linearly independent, the equation $$c_1v_1 + c_2v_2 \cdots + c_kv_k = 0$$ holds only when $$c_1 = c_2 = c_k = 0$$.

Any help would be appreciated.

• If you have $n+1$ or more vectors in $\Bbb R^n$, they are guaranteed to be linearly dependent, no manual checking necessary. As for checking if the first three are independent of one another, surely you've seen at least one technique demonstrated before? Perhaps you've seen row reduction used for this? – JMoravitz Oct 22 '18 at 3:00
• In other words, set up a homogenous linear system with unknown coefficients $c_1,c_2,c_3$ for the first part of the problem to find out if a nontrivial solution (not all coefficients zero) is possible. See this introduction to posting math notation. – hardmath Oct 22 '18 at 3:05