## Linearly Independent/Dependent Vectors Question

## Problem 48

Let $V$ be an $n$-dimensional vector space over a field $K$.

Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$.

Are the following vectors linearly independent?

\[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, \quad \dots, \quad \mathbf{v}_{k-1}+\mathbf{v}_k, \quad \mathbf{v}_k+\mathbf{v}_1.\]

If it is linearly dependent, give a non-trivial linear combination of these vectors summing up to the zero vector.

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