 Solution

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/c/solution

Every output

u_{1} + u_{2}

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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