TITLE: The union of two spanning sets
QUESTION [1 upvotes]: Consider three vectors a,b,c they are linear independent with each other. Let P = span{a,b}, Q = span{b,c}. Is the union of P,Q = span{a,b,c}? If not what is it? Thanks

REPLY [0 votes]: No, take in $\mathbb{R}^3$ the two sets
$$P=span\{\vec{i},\vec{j}\}\text{ and } Q=span\{\vec{j},\vec{k}\}$$
Clearly $P$ Is the $xy-$ plane and $Q$ is the $yz-$ plane so their union is not $\mathbb{R}^3$, where as
$$span\{\vec{i},\vec{j},\vec{k}\}=\mathbb{R}^3$$
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