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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-i/1/a/i/solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 1 / a / i / Solution by

Codex  0 2026-10-25

The vectors are linearly independent if

\sum_{j = 1}^{n} a_{j} v_{j} = 0, a_{j} \in R ⟹ a_{1} = \dots = a_{n} = 0.

They span

V

if every

v \in V

is a linear combination

v = \sum_{j = 1}^{n} a_{j} v_{j}

with real coefficients. They form a basis of

V

if they are linearly independent and span

V

. Equivalently, every

v \in V

has exactly one such representation: existence is the spanning set condition, and subtracting two representations shows that uniqueness is precisely linear independence.

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