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I will now state the exercise:

"Prove the following statement. Let β be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset S1 of β such that S ∪ S1 is a basis for V."

My question regards a fact he uses in his solution which I cannot understand how he deduces. I will now describe his solution and explain which part confuses me:

He starts by considering the family $F$ of all linearly independent subsets of β such that their union with S is linearly independent. Then he proves by the maximal principle (as referenced in his book page 59) that $F$ has a maximal element, denoted S1.

He then goes on to state precicely that "By the maximality of $S_1$ we know that $S_1 \cup S$ generates $β$ "

I understand why that implies our desired result, but what I don't understand is why it is true.

How do we know that $S_1 \cup S$ generates $β$ ?

Thank you in advance, any help is very much appreciated

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So assume for contradiction that $S_1\cup S$ does not generate $\beta$. Then in particular, there is some $v\in\beta$ not generated by $S_1\cup S$. You can then check that this implies by definition that $S_1\cup S\cup\{v\}$ is also independent. Thus $S_1$ is not maximal as it is bounded above by $S_1\subsetneq S_1\cup\{v\}\subseteq \beta$.

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    $\begingroup$ Thank you very much, gonna have to keep that in mind when choosing families of functions for similar exercises, +1 $\endgroup$
    – Stamatis
    Jan 6, 2021 at 22:07

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