In Problems 21–24 verify that the
24.
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Chapter 8 Solutions
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- 8arrow_forward2. Given the following 2 x 2 linear system with constant coefficients x' = Ax (H) x= Ax+g(t), (N) where g is not the zero vector. Which of the following statements are true? Justify your answers. A. If , is a solution to (H) and 7, is a solution to (N), then , +27, is a solution to (N). B. If , and 2 are both solutions to (N), then ₁-2 is a solution to (H).arrow_forward14. Assume x E R. Give the matrix associated with the quadratic form 10(x,) - 5x,x2 + 6(x,).arrow_forward
- In Problem. -24, the given vector functions are solutions to a system x' (t) = Ax(t). Determine whether they form a fundamental solution set. If they do, find a fundamental matrix for the system and give a general solution. -E]. et 24. x₁ = X2 sint COS [ -sint_ X cost sin/ COSTarrow_forward10. Find the general solution of the system of differential equations 3 -2 -2 d. X = -3 -2 -6 X dt 3 10 1 + 2tet + 3t?et + 4t°et 3 1 -3 Hint: The characteristic polymomial of the coefficient matrix is -(A- 4)²(A- 3). Moreover (:) 2 1 Xp(t) = t²et +t³et +t'e3t -1 -1 -3 is a particular solution of the system.arrow_forward2. factor 1/4, then reflects about the line y = x. 01 1/4 0 0 1/4 (a) A = (d) A = (₁ Find the standard matrix for the operator on R² which contracts with ( :) :) 1/4 0 0 1/4 (1¹4) (e) None of these (b) A = 0 1 -(;!) 0 (c) A =arrow_forward
- In Exercises 11–14, find parametric equations for all least squares solutions of Ax = b, and confirm that all of the solutions have the same error vector. 1 3 1 12. A = -2 -6 |; b = ! 0 3 9. 1arrow_forward6. Find the general solution of the system of differen- tial equations 6 -1 1 1 d 6t X - e 6. -1 1 2 dt 1 1 3 Hint: The characteristic polynomial of the coefficient matrix is –(A – 7)²(A – 4). | 2.arrow_forwardWhat can you conclude about the values of the quadratic form Q(x)?arrow_forward
- 7. Consider the invertible matrix It is given that A-1 = b11 b21 b12 b22 2 -- (EE) b31 b32 1 2 -1 -- (7) = 2 -2 A 0 1 1 (a) Find the entry b21 of A-¹ using the adjoint formula. X (b) Solve the linear system AX + 2B = 0, where X = y B = " and 0 is the zero Z matrix of the appropriate size.arrow_forwardProblem 16 (#2.3.34).Let f(x) = ax +b, and g(x) = cx +d. Find a condition on the constants a, b, c, d such that f◦g=g◦f. Proof. By definition, f◦g(x) = a(cx +d) + b=acx +ad +b, and g◦f(x) = c(ax +b) + d=acx +bc +d. Setting the two equal, we see acx +ad +b=acx +bc +d ad +b=bc +d (a−1)d=(c−1)b That last step was merely added for aesthetic reasons.arrow_forward1. 2. 3. For which values of a and b is the following system of equations inconsistent. x+2y3z = 4 3x = y + 5z = 2 4x + y + az = b (a) a= 2 and b = 6; (d) a = 1 and b = 3; (d) A = Find the standard matrix for the operator on R² which contracts with factor 1/4, then reflects about the line y = x. 0 (a) A = 1/4 0 (₁/11) ( 0 1/4 1/4 ¹/4) 0 (b) a = 2 and b = 6; (e) None of these. (c) a 2 and b = 6; 0 1/4 - (¹/4) 0 (e) None of these (b) A = (e) None of these (c) A = The linear operator T : R³ → R³ is defined by T(x₁, x2, X3) = (W₁, W2, W3), where w₁ = 2x₁ + 4x2 + x3; W₂ = 9x2+2x3; W3 = 2x1 8x2 - 2x3. Which of the following is correct. (a) T is not one to one. (b) T is one to one but the standard matrix for T-¹ does not exist. (c) T is one to one and its standard matrix for T-¹ is (d) T is one to one and its standard matrix for T-¹ is HOLI 0 1 (88) 0 3 0 1 3 3 WIN - WIN 3 0 1 3 1 -4 3 -3 2 -3 1623arrow_forward
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