Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 1 / c

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- Solution c

Solution

Subtracting the first vector from each of the other two preserves the basis property by iii. The resulting family is

e_{1}, (0, α, α^{2})^{T}, (0, β, β^{2})^{T} .

Its last two vectors belong to the plane with first coordinate zero. By b, they are linearly independent there precisely when

α β^{2} - β α^{2} = α β (β - α) \neq = 0.

In that case their two-dimensional span is the whole plane, and adjoining

e_{1}

gives a basis of

R^{3}

. If the displayed expression vanishes, a nontrivial linear relation among those two vectors prevents a basis. Thus

α \neq = 0, β \neq = 0, α \neq = β .

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