(Chapter 4.2) Questions 8, 11, 13, 15, 21 No plagrism and please write on paper please thanks a lot

4.2 Subspaces 219 In Exercises 7-8, use the Subspace Test to determine which of the sets are subspaces of F(-co, Do) 17. (Calculus Required) Which of the following are subspaces of R? 7. a. All functions f in F(-00, co) for which f(0) = 0. a. All sequences of the form v = (U1, U2, . . ., Un, . . . ) such that b. All functions f in F(-oo, co) for which f(0) = 1. lim Un = 0. 8. 12 - +00 a. All functions f in F(-09, co) for which f(-x) = f(x). b. All convergent sequences (that is, all sequences of the form b. All polynomials of degree 2. v = (U1 , Uz, . . . , Un. . . . ) such that lim Up, exists). In Exercises 9-10, use the Subspace Test to determine which of the sets are subspaces of Roe. c. All sequences of the form v = (U1 , Uz, . . .. Un, . .. ) such that LinzI Un = 0. 9. a. All sequences v in Re of the form v = (v, 0, v, 0, v, 0, ...). d. All sequences of the form v = (U1 , U2. . .., Un, . . .) such that b. All sequences v in Roe of the form v = (v, 1, v, 1, v, 1, ... ) Linzl Un converges. 10. a. All sequences v in R. of the form 18. A line L through the origin in R' can be represented by para- metric equations of the form x = at, y = bt, and z = ct. Use V = (U, 20, 40, 80, 160, ...) these equations to show that L is a subspace of R* by showing b. All sequences in Ro whose components are 0 from some that ifv, = (x1 , )1, Z ) and v2 = (Xz, )/2, Zz) are points on Land point on. k is any real number, then kv, and v, + V, are also points on L. In Exercises 11-12, use the Subspace Test to determine which of the 19. Determine whether the solution space of the system Ax = 0 sets are subspaces of M 22- is a line through the origin, a plane through the origin, or the origin only. If it is a plane, find an equation for it. If it is a line, 11. a. All matrices of the form find parametric equations for it. 2 3 b. All matrices of the form b 1- a. A = 3 0 b. A = 2 5 3 2 - 5 0 8 c. All 2 X 2 matrices A such that [-1]= 13] C. A = 2 2 d. A = 2 - 1 4 - 9 3 1 11 12. a. All 2 X 2 matrices A such that 20. (Calculus required) Show that the following sets of functions [-1]= 18] are subspaces of F(-co, co) b. All 2 x 2 matrices A such that a. All continuous functions on (-co, co). b. All differentiable functions on (-co, co). c. All differentiable functions on (-co, co ) that satisfy f' + 2f = 0. c. All 2 x 2 matrices A for which det(A) = 0. In Exercises 13-14, use the Subspace Test to determine which of the 21. (Calculus required) Show that the set of continuous func- tions f = f(x) on [a, b] such that sets are subspaces of R*. 13. a. All vectors of the form (a, a2, a], at). b. All vectors of the form (a, 0, b, 0). is a subspace of C[a, b]. 14. a. All vectors x in R4 such that Ax = , where 22. Show that the solution vectors of a consistent nonhomoge- A =_ - 1 0 2 1 neous system of m linear equations in n unknowns do not form a subspace of R". 23. If TA is multiplication by a matrix A with three columns, then b. All vectors x in R4 such that Ax = 1, where A is as in the kernel of TA is one of four possible geometric objects. What part (a). are they? Explain how you reached your conclusion. In Exercises 15-16, use the Subspace Test to determine which of the 24. Consider the following subsets of P3: V consists of all poly- sets are subspaces of P... nomials do + ax + ax + aye such that do + a; = 0 and W 15. a. All polynomials of degree less than or equal to 6. consists of all polynomials p such that p(1) = 0. b. All polynomials of degree equal to 6. . Use the Subspace Test to show that V and W are subspaces of P3. c. All polynomials of degree greater than or equal to 6. b. Show that the set of all polynomials 16. a. All polynomials with even coefficients. b. All polynomials whose coefficients sum to 0. P = do tax+ azx2+ a3x5 c. All polynomials of even degree. such that do + a; = 0 and p(1) = 0 is a subspace of P, with- out using the Subspace Test