(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

After trying a two separate (and somewhat lengthy) approaches, both yielded that the dimension of W is n, and [tex]e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}, e_{n}-e_{1}[/tex] 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 [tex]e_{1}-e_{2}, e_{2}-e_{3}, ..., e_{n-1}-e_{n}[/tex].

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

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# Homework Help: Dimension and basis for subspace determined by given vectors

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