Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 5 / d

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- Solution d

Solution

Expand the quadratic form of this Gram matrix:

Q (c) = i, j \sum c_{i} c_{j} x_{i}^{T} x_{j} = (i \sum c_{i} x_{i})^{T} (j \sum c_{j} x_{j}) = i \sum c_{i} x_{i}^{2} .

For

c \neq = 0

, linear independence of the real vectors implies

\sum_{i} c_{i} x_{i} \neq = 0

, so its squared Euclidean norm is strictly positive. Thus

Q (c) > 0 for every c \neq = 0,

proving the positive definiteness of a Gram matrix in the independent-vector case.

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