In Problems 13-19, determine whether the given
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Fundamentals of Differential Equations and Boundary Value Problems
- Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.arrow_forwardLet f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].arrow_forwardIf the vector function F = a (3y – k1 z) + a (k2x – 2z) - az (k3y + z) is irrotational, then the values of the constants k1, k2 and k3, respectively, arearrow_forward
- [1 4] 2. Let D = 2 5 3 6 [4 3] E = 5 6 6 9 a = {B b = {n Suppose now a vector z = {9 + 0 satisfies D'z = 0. Answer the following questions. (1) Find the relations between p, q and r. (2) When some solution vector x = exists to the equation Dx = a, show the necessary condition that a, B and y should satisfy. (3) When some solution vector y = exists to the equation Ey = b, show the necessary condition that 5, 7 and x should satisfy.arrow_forward1) Find the domain of the vector-valued function r(t) = (3t+1,6,ln(9-t)).arrow_forwardEvaluate the vector-valued function at each given value of t. (If you need to use At, enter Deltat.) r(t) = i - (t - 4)j (a) r(4) = 8 (b) r(0) (c) r(s+ 4) = (ar - 4at)i- 2aj (d) r(2 + At) - r(2) =arrow_forward
- The vectors -4 [] [ 11 V= 30 are linearly independent if and only if k ū= 4 —8 -17+k 1 [3] 1 -4arrow_forwardProblem 3. Determine whether the following polynomials are linearly independent in the vector space P of polynomials: (i) 2, x², x and 2x + 3; (ii) x + 2, x + 1 and 2² 1.arrow_forwardSuppose S = {r, u, d} is a set of linearly independent vectors. If x = 3r+ 5u + 2d, determine whether T = {r, u, x} is a linearly independent set. Select an Answer 1. Is I linearly independent or dependent? If I' is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. r+ u+ x = 0.arrow_forward
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