# Linear Algebra Friedberg Exercise 1.7.7 Query

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

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 $$S_1\subsetneq S_1\cup\{v\}\subseteq \beta$$.