 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/digital-systems-linear-algebra-introduction-to-formal-proof/6/b/solution

v_{1}, \dots, v_{n}

α_{1} v_{1} + \dots + α_{n} v_{n} = 0 ⟹ α_{1} = \dots = α_{n} = 0.

They are linearly dependent if some such relation has at least one nonzero coefficient. All coefficients lie in the underlying field. Thus linear independence means that only the trivial linear combination gives zero; linear dependence means that a nontrivial one does.

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