Vector Space Problem: Maximal Linear Independence & Minimal Spanning Set

[ak123456]

Homework Statement

Let V be a vector space over a field F and let x1,...xn $$\in$$V.Suppose that x1,...xn form a maximal linearly independent subset of V. Show that x1,...xn form a minimal spanning set of V.

Homework Equations

The Attempt at a Solution

I knew that x1,...xn are linear independent and for every x$$\in$$V the n+1 vectors x1,...xn , x are linear dependent
then x span x1,...xn
i don't how to continue
any help ?

 

[lanedance]

Homework Statement

Let V be a vector space over a field F and let x1,...xn $$\in$$V.Suppose that x1,...xn form a maximal linearly independent subset of V. Show that x1,...xn form a minimal spanning set of V.

Homework Equations

The Attempt at a Solution

I knew that x1,...xn are linear independent and for every x$$\in$$V the n+1 vectors x1,...xn , x are linear dependent
then x span x1,...xn
i don't how to continue
any help ?

hey ak123456

first you have only n vectors

what are your defintions for maximally lineraly independent and minimal span? always a good place to start

to get you started though, the my reasoning would be as follows:
as {x1,...,xn} is maximal lineraly independent show any vector in V can be written as combination of xn's, so {x1,...,xn} spans V.
then try and show if you remove an xi, the remaining vectors no longer span V...

 

[HallsofIvy]

If {x1, ..., xn} did not span the set, there must exist some x which cannot be written as a combination of the {x1, ..., xn}. What does that tell you about {x1, x2, ..., xn, x}?

If it were not a minimal spanning set, then there must exist a smaller set, {y1, y2,... yn-1} which did span the set. That would mean you could write each of x1, x2, ... xn in terms of those y's. What does that tell you about the independence of x1, x2, ..., xn?