4.1 Exercises 1. Let V be the first quadrant in the xy-plane; that is, let 7. All polynomials of degree at most 3, with integers as coeffi- cients. 8. All polynomials in P,, such that p(0) = 0. a. If u and v are in V, is u + vin V? Why? b. Find a specific vector u in V and a specific scalar c such that cu is not in V. (This is enough to show that V is not 9. Let / be the set of all vectors of the form 3.s Find a a vector space.) 2.s vector v in Ro such that H = Span {v). Why does this show 2. Let W be the union of the first and third quadrants in the xy- that H is a subspace of IR3? plane. That is, let W a. If u is in W and c is any scalar, is cu in W? Why? 10. Let // be the set of all vectors of the form ION . Show that b. Find specific vectors u and v in W such that u + v is not H is a subspace of R'. (Use the method of Exercise 9.) in W. (This is enough to show that W is not a vector space.) 5h + 2c 11. Let W be the set of all vectors of the form h 3. Let // be the set of points inside and on the unit circle in C the xy-plane. That is, let H = ( [ ] ty . Find where b and c are arbitrary. Find vectors u and v such that W = Span {u, v}. Why does this show that W is a subspace a specific example-two vectors or a vector and a scalar-to of R 3? show that // is not a subspace of IR2. S + 31 4. Construct a geometric figure that illustrates why a line in RR2 not through the origin is not closed under vector addition. 12. Let W be the set of all vectors of the form S' -- 1. . Show 2.5 - 1 In Exercises 5-8, determine if the given set is a subspace of P,, for 41 an appropriate value of n. Justify your answers. that W is a subspace of R4. (Use the method of Exercise 1 1.) 5. All polynomials of the form p(t) = at2, where a is in R. 6. All polynomials of the form p(t) = a + 2, where a is in R. 13. Let vi - 9 .2- - - 87.now-436. Fillin the missing axiom numbers in the following proof that 37. 39 f) = 0 for every scalar . c0=c(0+0) by Axiom (a) =0+ c0 by Axiom (b) Add the negative of 0 to both sides: 0+ (c0) = [e0+ O] + (ch) 0+ (=c0) = 0+ [0 4 (c0)] by Axiom (c) 0=c0+0 by Axiom (d) 0= c0 by Axiom (e) Prove that (1)u = u. [Hint: Show that u + (1)u = 0. Use some axioms and the results of Exercises 34 and 35.] Suppose cu = 0 for soine nonzero scalar . Show that u = 0. Mention the axioms or properties you use. Let u and v be vectors in a vector space V, and let /f be any subspace of that contains both u and v. Explain why I also contains Span {u, v}. This shows that Span {u, v} is the smallest subspace of that contains both w and v. 42, i 43. il 44. b. Show that 1 is a subspace of H + K and K is a subsp, of H + K. Suppose uy.....u, and vi,..., v, are vectors in a veg space V', and let H = Span{uy,...,u,} and K = Span{v,,...,v,} Show that // + K = Span{uy,... ,u,,.v|..._..'_v,,}, Show that w is in the subspace of B* spanned by v, vy, where 9 8 4 iz ws | | s o e | R 1o G =3 i v r% y 12 ) v = g 7 9 8 18 Determine if y is in the subspace of R* spanned by columns of A, where 4 g =8 -8 R 7 -6 P gfx 4=V 5 g 3 esff B =3 =5