Let S be a spanning set for a finite dimensional vector space V. Prove that there exists a subset S′ of S that forms a basis for V.Getting Started: S is a spanning set, but it may not be a basis because it may be linearly dependent. You need to remove extra vectors so that a subset S′ is a spanning set and is also linearly independent.(i) If S is a linearly independent set, then you are done. If not, remove some vector v from S that is a linear combination of the other vectors in S. Call this set S1.(ii) If S1 is a linearly independent set, then you are done. If not, then continue to remove dependent vectors until you produce a linearly independent subset S′.(iii) Conclude that this subset is the minimal spanning set S′.

Let S be a spanning set for a finite dimensional vector space V. Prove that there exists a subset S′ of S that forms a basis for V.
Getting Started: S is a spanning set, but it may not be a basis because it may be linearly dependent. You need to remove extra vectors so that a subset S′ is a spanning set and is also linearly independent.
(i) If S is a linearly independent set, then you are done. If not, remove some vector v from S that is a linear combination of the other vectors in S. Call this set S₁.
(ii) If S₁ is a linearly independent set, then you are done. If not, then continue to remove dependent vectors until you produce a linearly independent subset S′.
(iii) Conclude that this subset is the minimal spanning set S′.