 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit/9/a/solution

F

\sum_{j = 1}^{n} c_{j} v_{j} = 0

with

c_{j} \in F

implies

c_{j} = 0

for every

j

. Otherwise there is a nontrivial linear dependence. Forming a matrix with the displayed vectors as columns gives

det α 32 240 - 1 04 = 16 α - 16.

Thus

α = 1

is the unique dependent case. At that value the explicit relation

2 v_{1} - \frac{3}{2} v_{2} - v_{3} = 0

verifies the dependence.

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