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Linear Algebra

Exercise 5.4.11

Answers

1.

Let

w

be an element in

W

. We may express

w

to be

w = \sum_{i = 0}^{k} a_{i} T^{i} (v) .

And thus we have

T (w) = \sum_{i = 0}^{k} a_{i} T^{i + 1} (v) \in W .

2.

Let

U

be a

T

-invariant subspace of

V

containing

v

. Since it’s

T

-invariant, we know that

T (v)

is an element in

U

. Inductively, we know that

T^{k} (v) \in U

for all nonnegative integer

k

. By Theorem 1.5 we know that

U

must contain

W

.

jephianlin

2011-06-27 00:00

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