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0 $begingroup$ I wrote a proof for a theorem and would appreciate if someone could check if my proof is sound. Thanks a lot. $Theorem$ : Every set that contains a linearly dependent set is linearly dependent. $Proof$ : Let $S={v_1, v_2,...,v_k}$ be a set of vectors. Let $L={v_1, v_2, ..., v_l} subset S$ , $l lt k$ , be a linearly dependent subset of $S$ . Since $L$ is linearly dependent, there must exist, without loss of generality, $v_1 in L$ such that $$sum_{i=2}^l alpha_i v_i = v_1, alpha_i in mathbb{R}$$ Then we can write a sum $$sum_{i=2}^l alpha_i v_i + sum_{i = l+1}^k 0 v_i= v_1 =$$ $$sum_{i=2}^k alpha_i v_i = v_1$$ where $alpha_i=0$ for $l+1 lt ilt k$ . Rearranging, $$sum_{i=2}^k alpha_i v_i - v_1 = 0 iff sum_{i=1}^k alpha_i v_i = 0$$ with $alpha_i in mathbb{R}$ not all zero.