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$S_1$ and $S_2$ finite sets on $\mathbb{R}^n$, $S_1$ is a subset of $S_2$, $(S_1\neq S_2)$. If $S_2$ is linearly dependent, so:

$S_1$ could be linearly dependent?

$S_1$ Could be linearly independent?

How can i answer this, and show some examples?

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up vote 1 down vote accepted

You can have both of them. For example suppose $S_{2}=\lbrace e_{1} , \ldots, e_{n} , e_{1} + e_{2} \rbrace$. If you have $S_{1}= \lbrace e_{1} , \ldots , e_{n} \rbrace$, it is linear independent. On the other hand you can have $S_{1} = \lbrace e_{1} , e_{2} , e_{1} + e_{2} \rbrace$. So there is no generic answer.

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