 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2012/1/b/solution

W^{0} = {λ \in V^{'} : λ (w) = 0 for every w \in W}, dim W^{0} = dim V - dim W .

For the dimension identity, choose a basis

w_{1}, \dots, w_{k}

W

and use basis extension to obtain

w_{1}, \dots, w_{n}

for

V

. A linear functional is determined by its values on these

n

k

values to zero and leaves its last

n - k

values arbitrary. Equivalently the last

n - k

elements of the dual basis form a basis of

W^{0}

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