Dimension and basis for subspace determined by given vectors

[Combinatus]

Homework Statement

Assume that e_1 ,..., e_n is a basis for the vector space V. Let W be the linear subspace determined (formed?) by the vectors e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1}. Determine the dimension of W, and a basis for W.

Homework Equations

The Attempt at a Solution

After trying a two separate (and somewhat lengthy) approaches, both yielded that the dimension of W is n, and e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1} forms the basis for W, i.e. no manipulation needed since the aforementioned vectors should already be linearly independent.

The key to the problem states that the subspace states that the dimension should rather be n-1, and the basis e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}.

After considering the key applied to a 3D vector space with the basis e_1, e_2, e_3, the key makes sense, since e_3-e_1 will be parallel to the plane formed by e_1-e_2 and e_2-e_3. I'm not certain how I should apply this knowledge to n-dimensional space.

 

[Combinatus]

I figured it out. Basically, I showed that the vectors e_1-e_2, e_2-e_3, ... , e_n-e_1 will for any dimension be linearly dependent (since the vector sum of all of them will point back to the origin), and excluding one made it possible to show that the new set of vectors will be linearly dependent. It was a reasonably entertaining problem though (which wasn't too hard after all), so if you want to give it a shot, or try a different approach, go for it!

 

[HallsofIvy]

Another way to do this is to note that the matrix having the new vectors as columns has all "1"s along the main diagonal and "-1" just below the main diagonal. Then it's easy to show that this matrix is non-singular so the vectors are independent. But your method is perfectly good.