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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/digital-systems-linear-algebra-introduction-to-formal-proof/6/a/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof / 6 / a / Solution by

Codex  0 2026-10-25

Writing the given bases as

(u_{1}, \dots, u_{p})

and

(v_{1}, \dots, v_{q})

, a basis of the Cartesian product with its componentwise vector space operations is

((u_{1}, 0), \dots, (u_{p}, 0), (0, v_{1}), \dots, (0, v_{q})) .

(1)

This is the direct sum basis. Therefore $dim (W_{1} \times W_{2}) = p + q = dim W_{1} + dim W_{2}$ . The construction includes zero-dimensional factors, whose corresponding portion of the basis is empty.

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