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 Incoming links: Direct sum basis

Addition map of subspaces 2026-10-25

For vector subspaces

U, W

of a common vector space, this is a surjective linear map. Its kernel of a linear map is

{(x, - x) : x \in U \cap W}

, which has a linear isomorphism to

U \cap W

by

x \mapsto (x, - x)

. The rank-nullity theorem and the direct sum basis give the dimension formula for a sum of subspaces. The paired domain keeps the two contributions separate even when their underlying vectors overlap.

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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 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})) .

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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