# Homework Help: Linear independance

1. Feb 21, 2006

### 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 dependant 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 independant

2. Feb 22, 2006

### 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?

3. Feb 22, 2006

### EnumaElish

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

Last edited: Feb 22, 2006