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Exercise 7.2.13

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1.

T^{i} (x) = 0

, then

T^{i + 1} (x) = T^{i} (T (x)) = 0

2.

Pick

β_{1}

to be one arbitrary basis for

N (T^{1})

. Extend

β_{i}

to be a basis for

N (T^{i + 1})

. Doing this inductively, we get the described sequence.

3.

By Exercise 7.1.7(c), we know

N (T^{i}) \neq N (T^{i + 1})

for

i \leq p - 1

. And the desired result comes from the fact that

T (β_{i}) \subset N (T^{i - 1}) = span (β_{i - 1}) \neq span (β_{i}) .

4.

The form of the characteristic polynomial directly comes from the previous argument. And the other observation is natural if the characteristic polynomial has been fixed.

jephianlin

2011-06-27 00:00

Exercise 7.2.13

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