Question

Plesae help with thia linear algebra question

(20) Let V be a vector space over the field K. Prove that if S is a linearly independent subset of V, then there exists a basis of V that contains S

Answer 1

Defination : A subset S of V is said to be a basis of v if and only if S is a linearly independen set and S spans V .

Now V is a vector space over a field K and S is a linearly independent subset of V .

If span (S) = V then by the above defination S forms a basis of V and so S contains in a basis is true .

But if span (S) $\neq$ V , then there exist v₁$\in$ V\ S such that S₁ = S $\cup$ { v₁ } is a linearly independent set .

If span ( S₁ ) = V then S₁ forms a basis of v and S₁ contains S so S contains in a basis of V .

But if span (S₁) $\neq$ V then there exist v₂$\in$ V\ S₁ such that S₁ = S $\cup$ { v₁ , v₂ } is a linearly independent set .

If span ( S₂ ) = V then S₂ forms a basis of v and S₂ contains S so S contains in a basis of V .

Proceeding this way we can find a basis of V such that S contained in that basis .

N.B. This is so called EXTENSION LEMMA .

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