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 / c

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- Solution c

Solution

Every output

u_{1} + u_{2}

lies in the sum of subspaces

W_{1} + W_{2}

, since it is a linear combination of members of the two vector subspaces. For

u = (u_{1}, u_{2})

,

v = (v_{1}, v_{2})

a, b

,

g (a u + b v) = (a u_{1} + b v_{1}) + (a u_{2} + b v_{2}) = a (u_{1} + u_{2}) + b (v_{1} + v_{2}) = a g (u) + b g (v) .

Therefore $g$ is a linear map. Every vector in

W_{1} + W_{2}

is such an output: group the terms of any linear combination from

W_{1} \cup W_{2}

according to their originating vector subspace. Thus the addition map of subspaces is also surjective.

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