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Homework 1

Exercise 1.A.12. (Notice that this is similar to #5, but the vector space is Fⁿ, where F is a field, like R or C. So the elements are n-tuples with entries in the field.
Exercise 1.B.3

Homework 2

Exercise 1.C.6. Hint: One is and one is not. Also, the key to one of them (at least the way I proved it) is to find an a and b such that a ≠ b but a³=b³.
Exercise 1.C.12

Homework 3

Exercise 2.A.3. Make sure you justify/prove your choice of t.
Exercise 2.A.7

Homework 4

Exercise 2.B.4

Homework 5

Exercise 2.C.10.
Hint: Use induction. Also, given a polynomial p_j of degree j, write it as p_j = a_jx^j+q_j-1(x), where a_j ≠ 0 and q_j-1(x) is some polynomial of degree at most j-1.

Homework 6

Exercise 3.A.5. (Show all of the properties from Definition 1.19.)

Homework 7

Exercise 3.B.21.

Homework 8

Exercise 3.C.15

Homework 9

Exercise 3.D.13

Homework 10

Exercise 5.A.12

Homework 11

Exercise 5.A.31

Homework 12

Exercise 5.C.16 a-d

Homework 13

Exercise 6.A.20.

Homework 14

Exercise 6.A.31.

Homework 15

Exercise 6.B.8.

Homework 16

Exercise 6.B.14.

Homework 17

Exercise 6.C.10 (Hint: I used Exercises 6.C.5 and 6.C.9)

Homework 18

Exercise 7.A.4

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Homework 19

Exercise 7.A.14
Exercise 7.A.17.
Hints for this one (Hopefully I haven't made any typos):
For the first part, start by showing that null T ⊂ null T^k, which is fairly straightforward. Showing that null T^k ⊂ null T takes a little more effort. One way is to show that T^*T^k-1v=0 for any v ∊ null T^k (do so by showing its length is 0, for instance). Then use that and normality to show that ⟨T^k-1v,T^k-1v⟩=0 (what does that imply?). Then argue that this eventually leads to v ∊ null T.
For the second part, show that range T^k ⊂ range T and then show they have the same dimension (why is that enough?).

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Homework 20

Exercise 7.B.11.

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Homework 21

Exercise 7.C.10.

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Homework 22

Exercise 7.D.13. Make sure to justify each step of your proof!

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Homework 23

Exercise 8.B.7. (Hint: mimic 8.31/8.33. Provide enough details so that it is clear to me that the details are clear to you.)

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Homework 24

Exercise 8.C.9.

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