 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2011/2/b/solution

A real inner product must be symmetric. Here

β_{a} (e_{1}, e_{2}) = - 2

and

β_{a} (e_{2}, e_{1}) = - a

a = 2

. For this value,

β_{2} (x, x) = 4 x_{1}^{2} - 4 x_{1} x_{2} + 3 x_{2}^{2} = 4 (x_{1} - x_{2} /2)^{2} + 2 x_{2}^{2} .

This is strictly positive for every nonzero vector. Bilinearity is already given, so symmetry and positive definiteness prove

a = 2 is the unique admissible value .

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