Showing that something is a basis of an independent set

[jumbogala]

Homework Statement

Among all independent vector sets in a vector space U, let M = {v1, v2, ... vp} be an independent set. p is as large as it can get. Show that U is a basis of M.

Homework Equations

The Attempt at a Solution

If U is a basis of M then U is an independent set (we already know it is) and U spans M.

Or, since the dimension is the maximum number of linearly independent vectors you can have in a subset, if dim(U) = the number of elements in M, then it is a basis.

dim(U) = p, since p is as big as it can get
and there are p elements in M so it's a basis.

That seems too simple though. Plus it doesn't show that U spans M, which I think is probably necessary. Can anyone help?

 

[vela]

I don't think you're supposed to use the concept of dimension yet. You also seemed to switch the role of U and M midstream.

Let's call the vector space U and the collection of independent vectors M. As you noted, you need to show that the vectors in M are linearly independent and that they span U. The first condition is true by assumption, so you just need to show M spans U. I suggest you assume M doesn't span U and show it leads to a contradiction.