10. The Proof of Theorem 6.21. Use the ideas from Exercise (9) to prove The- orem 6.21. Let A, B, and C be nonempty sets and let f: A →B and g: B C. (a) If go f: A -→ C is an injection, then f: A → B is an injection. (b) If go f: A → C is a surjection, then g: B → C is a surjection. Hint: For part (a), start by asking, "What do we have to do to prove that f is an injection?" Start with a similar question for part (b).

10. The Proof of Theorem 6.21. Use the ideas from Exercise (9) to prove The- orem 6.21. Let A, B, and C be nonempty sets and let f: A →B and g: B C. (a) If go f: A -→ C is an injection, then f: A → B is an injection. (b) If go f: A → C is a surjection, then g: B → C is a surjection. Hint: For part (a), start by asking, "What do we have to do to prove that f is an injection?" Start with a similar question for part (b).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

Problem 9BEXP

See similar textbooks

Similar questions

18. Let and be defined as follows. In each case, compute for arbitrary . a. b. c. d. e.
Let f:AA, where A is nonempty. Prove that f a has right inverse if and only if f(f1(T))=T for every subset T of A.
25. Let, where and are non empty, and let and be subsets of . Prove that. Prove that. Prove that. Prove that if.
For the given f:ZZ, decide whether f is onto and whether it is one-to-one. Prove that your decisions are correct. a. f(x)={ x2ifxiseven0ifxisodd b. f(x)={ 0ifxiseven2xifxisodd c. f(x)={ 2x+1ifxisevenx+12ifxisodd d. f(x)={ x2ifxisevenx32ifxisodd e. f(x)={ 3xifxiseven2xifxisodd f. f(x)={ 2x1ifxiseven2xifxisodd
27. Let , where and are nonempty. Prove that has the property that for every subset of if and only if is one-to-one. (Compare with Exercise 15 b.). 15. b. For the mapping , show that if , then .
4. Let , where is nonempty. Prove that a has left inverse if and only if for every subset of .
24. Let, where and are nonempty. Prove that for all subsets and of. Prove that for all subsets and of. Give an example where there are subsets and of such that. Prove that for all subsets and of. Give an example where there are subsets and of such that .