4.1 Questions 10, 17, 21, 22 No plagrism and please write on paper thanks a lot
d. Show that Axiom 5 holds by producing a vector -u such that u + (-u) = 0 for u = (u,, uz) e. Find two vector space axioms that fail to hold. In Exercises 3-12, determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. 3. The set of all real numbers with the standard operations of addition and multiplication. 4. The set of all pairs of real numbers of the form (x, 0) with the standard operations on R2. 5. The set of all pairs of real numbers of the form (x, y), where x 2 0, with the standard operations on R2. 6. The set of all n-tuples of real numbers that have the form (x, X, . . .,x) with the standard operations on Rn. 7. The set of all triples of real numbers with the standard vector addition but with scalar multiplication defined by k (x, y, z) = ( k2x, kay, k=z) 8. The set of all 2 x 2 invertible matrices with the standard matrix addition and scalar multiplication. 9. The set of all 2 x 2 matrices of the form with the standard matrix addition and scalar multiplication. 210 CHAPTER 4 General Vector Spaces 10. The set of all real-valued functions f defined everywhere on the real line and such that f(1) = 0 with the operations used by specifying which of the ten vector space axioms applies in Example 6. Hypothesis: Let u be any vector in a vector space V, let 0 be the 11. The set of all pairs of real numbers of the form (1, x) with the zero vector in V, and let k be a scalar. operations Conclusion: Then ko = 0. (1,y) + (1,y') = (1,y+y') and k(1,y) = (1, ky) Proof: (1) ko + ku = k(0 + u) 12. The set of polynomials of the form a, + a x with the opera- (2) = ku tions (3) Since ku is in V, -ku is in V. ( do + a, x) + (bo + bix) = (do+ bo) + ( a, + b, )x (4) Therefore, (ko + ku) + (-ku) = ku + (-ku). and (5) ko + (ku + (-ku)) =ku+(-ku) k(do + a, x) = (kao) + (ka, )x (6) k0 +0 =0 (7) *0 = 0 13. Verify Axioms 3, 7, 8, and 9 for the vector space given in Exam- ple 4. 14. Verify Axioms 1, 2, 3, 7, 8, 9, and 10 for the vector space given In Exercises 23-24, let u be any vector in a vector space V. Give a step-by-step proof of the stated result using Exercises 21 and 22 as in Example 6. models for your presentation. 15. With the addition and scalar multiplication operations defined 23. Ou = 0 24. -u= (-1)u in Example 7, show that V = R2 satisfies Axioms 1-9. In Exercises 25-27, prove that the given set with the stated operations 16. Verify Axioms 1, 2, 3, 6, 8, 9, and 10 for the vector space given is a vector space. in Example 8. 25. The set V = {0} with the operations of addition and scalar 17. Show that the set of all points in R' lying on a line is a vector multiplication given in Example 1. space with respect to the standard operations of vector addi- ion and scalar multiplication if and only if the line passes 26. The set Ro of all infinite sequences of real numbers with through the origin. the operations of addition and scalar multiplication given in Example 3. 18. Show that the set of all points in R lying in a plane is a vector space with respect to the standard operations of vector addi- 27. The set Mmm of all m x n matrices with the usual operations of tion and scalar multiplication if and only if the plane passes addition and scalar multiplication. through the origin. 28. Prove: If u is a vector in a vector space V and k a scalar such In Exercises 19-20, let V be the vector space of positive real num- that ku = 0, then either k = 0 or u = 0. [Suggestion: Show bers with the vector space operations given in Example 8. Let u = u that if ku = 0 and k # 0, then u = 0. The result then follows be any vector in V, and rewrite the vector statement as a statement as a logical consequence of this.] about real numbers. 19. -u = (-1)u True-False Exercises 20. ku = 0 if and only if k = 0 or u = 0. TF. In parts (a)-(f) determine whether the statement is true or false, and justify your answer. Working with Proofs a. A vector is any element of a vector space. 21. The argument that follows proves that if u, v, and w are vec- tors in a vector space I such that u + w = v + w, then u = v b. A vector space must contain at least two vectors. (the cancellation law for vector addition). As illustrated, jus- c. If u is a vector and k is a scalar such that ku = 0, then it tify the steps by filling in the blanks. must be true that k = 0. utw= v+w Hypothesis (u + w) +(-w) = (v+w)+(-w) Add -w to both sides. d. The set of positive real numbers is a vector space if vec- tor addition and scalar multiplication are the usual oper- u+ [w+(-w)]= v+[w+(-w)] ations of addition and multiplication of real numbers. u+0 =v+0 1 = V e. In every vector space the vectors (-1)u and -u are the same. 22. The seven-step proof of part (b) of Theorem 4.1.1 follows. Jus- tify each step either by stating that it is true by hypothesis or f. In the vector space F(-co, co) any function whose graph passes through the origin is a zero vector
