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So I am stuck on a problem in my practice sheet (not homework) and I had a question regarding this problem.

Let V be a vector space, W a subspace of V , S a linearly independent subset of W , and v ∈ V \W.Prove that S ∪ {v} is linearly independent.

I wanted to know what is the meaning of the notation V\W? Does it have any effect on the way we solve the problem? I think it means that v belongs to V but not W

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2 Answers 2

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I cannot comment yet but I am just trying to help. Anyone can correct me if I'm wrong.

Since $S \subset W$ then Span$(S)$ is a subspace of $W$, and since it is given that $v \notin W$, then $v \notin$ Span$(S)$ and therefore no linear combination of $S$ could produce $v$ and so $S \ \cup \ \{v\}$ is linearly independent.

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    $\begingroup$ Use \notin to get $\notin$ $\endgroup$
    – egreg
    Apr 28, 2021 at 7:40
  • $\begingroup$ Thanks a lot! I got it now! $\endgroup$
    – Areen
    Apr 28, 2021 at 9:53
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$V \setminus W$ is $V$ without $W$ and is sometimes written $V-W$. The latex command for this is \setminus for this reason. Formally if $v \in V \setminus W$ then $v \in V, v \notin W$.

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  • $\begingroup$ Understood! Thanks a lot! $\endgroup$
    – Areen
    Apr 28, 2021 at 9:54

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