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DeEl howard-anton-chris-rorres-elementary-linear-algebra-applications-version-11th-edition.pdf X 4.2 Subspaces 201 (b) All sequences v in R" of the form origin only. If it is a plane, find an equation for it. If it is a v = (v, 1, v, 1, v, 1, . . .). line, find parametric equations for it. (c) All sequences v in R" of the form v = (v, 2v, 4v, 8v, 16v, . . .). (a) A= 3 - 1 o ( b) A = 2 5 3 (d) All sequences in R" whose components are 0 from some point on. 6. 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 (@) A = 2 these equations to show that L is a subspace of R' by showing that if vi = (x1, y1, ZI) and v2 = (X2, )2, Z2) are points on L and & is any real number, then kv, and v1 + v2 are also points 16. (Calculus required) Show that the following sets of functions on L. are subspaces of F(-00, Do). (a) All continuous functions on (-00, 00). 7. Which of the following are linear combinations of u = (0, -2, 2) and v = (1, 3, -1)? (b) All differentiable functions on (-20, oo). (a) (2, 2, 2) (b) (0, 4, 5) (c) (0, 0, 0) (c) All differentiable functions on (-00, oo) that satisfy f' + 2f = 0. 8. Express the following as linear combinations of u = (2, 1, 4), v = (1, -1, 3), and w = (3, 2, 5). 17. (Calculus required) Show that the set of continuous functions f = f(x) on [a, b] such that (a) (-9, -7, -15) (b) (6, 11, 6) (c) (0, 0, 0) 9. Which of the following are linear combinations of A= 2 9. B=[ -3. c=[ ] is a subspace of Cla, b]. 18. Show that the solution vectors of a consistent nonhomoge- neous system of m linear equations in n unknowns do not form a subspace of R" . 10. In each part express the vector as a linear combination of 19. In each part, let TA: R2 -> R? be multiplication by A, and P1 = 2 + x + 4x-, P2 = 1 - x+ 3x, and let u) = (1, 2) and u2 = (-1, 1). Determine whether the set P3 = 3 + 2x + 5x2. (TA(uj). TA(u2)) spans R. (a) -9-7x - 15x2 (b) 6+ 11x + 6.x2 (c) 0 (d) 7+ 8x + 9x2 () 4- 6 -4] 11. In each part, determine whether the vectors span R3. 20. In each part, let TA: R3 -> R? be multiplication by A, and let (a) vi = (2, 2, 2), v2 = (0, 0, 3), v3 = (0, 1, 1) uj = (0, 1, 1) and u2 = (2, -1, 1) and us = (1, 1, -2). De- (b) v1 = (2, -1, 3), v2 = (4, 1, 2), v3 = (8, -1, 8) termine whether the set {TA (uj), TA(u2), TA(u;)) spans R2. 12. Suppose that v1 = (2, 1, 0, 3), v2 = (3, -1, 5, 2), and 3 = (-1, 0, 2, 1). Which of the following vectors are in (@)4= ! ! (6)A = 1 1 3] span (v1, v2, v3]? (a) (2, 3, -7, 3) (b) (0, 0, 0, 0) 21. If TA is multiplication by a matrix A with three columns, then the kernel of TA is one of four possible geometric objects. What (c) (1, 1, 1, 1) d) (-4, 6, -13, 4) are they? Explain how you reached your conclusion. 13. Determine whether the following polynomials span P2. 22. Let v1 = (1, 6, 4), v2 = (2, 4, -1), v3 = (-1, 2, 5), and P1 = 1 -x+2x2, P2 = 3+x, w/ = (1, -2, -5), w2 = (0, 8, 9). Use Theorem 4.2.6 to show p3 = 5 - x +4x, PA = -2 - 2x + 2x2 that span(v1, V2, V3] = span(w1, w2). 14. Let f = cos x and g = sin x. Which of the following lie in 23. The accompanying figure shows a mass-spring system in which the space spanned by f and g? a block of mass m is set into vibratory motion by pulling the block beyond its natural position at x = 0 and releasing it at (a) cos 2x (b) 3 + x2 (c) 1 (d) sinx (e) 0 time 1 = 0. If friction and air resistance are ignored, then the x-coordinate x (1) of the block at time t is given by a function 15. Determine whether the solution space of the system Ax = 0 of the form is a line through the origin, a plane through the origin, or the x (1) = ci cost + c2 sin wt Type here to search BORO S PDF PDF 32OF A DE 10:45 PM E 11/23/2021