 Solution

ID: past-exam-of-the-linear-algebra-course-of-the-university-of-oxford/2023/2/a/ii/solution

Put

d_{i} = dim K_{i}

, including

d_{0} = 0

T^{i}

K_{i + 1}

K_{i}

dim T^{i} (K_{i + 1}) = d_{i + 1} - d_{i} .

Likewise

T^{i - 1} ∣_{K_{i}}

K_{i - 1}

d_{i} - d_{i - 1}

. But

T (K_{i + 1}) \subseteq K_{i}

, which implies

T^{i} (K_{i + 1}) = T^{i - 1} (T (K_{i + 1})) \subseteq T^{i - 1} (K_{i}) .

d_{i + 1} - d_{i} \leq d_{i} - d_{i - 1}, 2 d_{i} \geq d_{i - 1} + d_{i + 1} (1 \leq i < m) .

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