 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-32/1/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Put

V = R^{d}

and write

a^{(k)}

for the degree-

k

T^{(N)} (V) = ⨁_{k = 0}^{N} V^{\otimes k}, (a \otimes b)^{(k)} = \sum_{j = 0}^{k} a^{(j)} \otimes b^{(k - j)} .

Here

V^{\otimes 0} = R

, and products of degree above

N

are discarded. The answers to the three printed parts follow with these conventions; the middle part is missing from the converted TeX.

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