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 / d

Table of contents
- Solution d

Solution

The kernel of a linear map

g

is precisely

U

, since

g (u_{1}, u_{2}) = 0 ⟺ u_{2} = - u_{1} ⟺ (u_{1}, u_{2}) = (u, - u), u \in W_{1} \cap W_{2} .

Part (c) gives

im g = W_{1} + W_{2}

. Apply the rank-nullity theorem to the finite-dimensional vector space

W_{1} \times W_{2}

, and then parts (a) and (b):

dim W_{1} + dim W_{2} = dim (W_{1} \times W_{2}) = dim ker g + dim im g = dim (W_{1} \cap W_{2}) + dim (W_{1} + W_{2}) .

This proves the dimension formula for a sum of subspaces. The kernel of a linear map records exactly the freedom to transfer a shared vector from one summand to the other without changing their sum.

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