Decide in this about linear independence. $$((1,0,2,1)^T,(2,0,1,1)^T,(1,0,1,−1)^T)⊂Q^4$$ How can I use this vectors to prove it? Can I use clasic analitics methods in linear algebra?
2026-03-28 14:25:45.1774707945
Linear independence in Q^4
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Take $a,b,c\in\mathbb Q$. Then\begin{align}a(1,0,2,1)^T+b(2,0,1,1)^T+c(1,0,1,-1)^T=(0,0,0,0)^T&\iff\left\{\begin{array}{l}a+2b+c=0\\0=0\\2a+b+c=0\\a+b-c=0\end{array}\right.\\&\iff\left\{\begin{array}{l}a+2b+c=0\\2a+b+c=0\\a+b-c=0.\end{array}\right.\end{align}This system is easy to solve. Its only solution is $(a,b,c)=(0,0,0)$. So, your vectors are linearly independent.