Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 3 / c / i

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- Solution i

Solution

Take any finite selection of distinct indices

i_{1}, \dots, i_{r}

and a relation

α_{1} e^{(i_{1})} + \dots + α_{r} e^{(i_{r})} = 0.

The coordinate at

i_{j}

on the left is exactly

α_{j}

, while the zero vector has coordinate zero. Hence

α_{j} = 0

for each

j

. Every finite subset of

E

is therefore linearly independent, which is the definition of linear independence for an infinite set. Thus $E$ is linearly independent in the vector space of finitely supported sequences.

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