Linear Independance: Show {v1,v2,...,vn} is Basis of V

[stunner5000pt]

Suppose that {v1,v2,...,vn} is a minimal spanning set for a vector space V. That is V = span {v1,v2,...,vn} and V cannot be spanned by fewer than n vectors. Show that {v1,v2,...,vn} is a basis of V

to be a basis for V then V = span (v1,v2,...,vn}
we already have that
suppose one of thise vectors was not linearly dependent then the number of vectors in the span is less than n. But V = span of n vectors
so the set of vectors must be lienarly dependent

 

[matt grime]

Surely the definition of a basis is that it is a minimal spanning set?

Or are you starting from linearly independent spanning set and showing any such thing is minimal, and vice versa?

 

[EnumaElish]

Suppose that {v1,v2,...,vn} is a minimal spanning set for a vector space V. That is V = span {v1,v2,...,vn} and V cannot be spanned by fewer than n vectors. Show that {v1,v2,...,vn} is a basis of V

to be a basis for V then V = span (v1,v2,...,vn}
we already have that
suppose one of thise vectors was not linearly dependent then the number of vectors in the span is less than n. But V = span of n vectors
so the set of vectors must be lienarly dependent

Yes, to be minimal, they have to be independent. Suppose not, then you could do away with at least one (how?) and still span V (you should work out an example here), which contradicts "minimal."

See http://en.wikipedia.org/wiki/Vector_basis