In Problems 13-19, determine whether the given
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Fundamentals of Differential Equations and Boundary Value Problems
- Let 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_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_forwardIf |r (t)| = c(a constant), which of the following is NOT true about the vector function r (t) O (t) and B (t) are orthogonal O T (t) and r (t) are orthogonal O T (t) and N (t) are orthogomal O (t) and r' (t) are orthogonalarrow_forward
- State whether the following derivatives of vector-valued functions rị(1) and r2(1) are scalars or vectors: d (a) ri(1) d (b) dt (r|(1) · r2(1) (c) (r|(1) x r2(1)) dt dtarrow_forwardShow that the vectors p1(x) = a + bx and p2(x) = c + dx are linearly independent in P1(R) if and only if the constants a, b, c, d satisfy ad − bc ≠ 0arrow_forwardwhich of the pollowing rets functions that contain are linearly independent ? C3x (3x -1 x+2, 4x) a. C4, 2x -3) (ex) C2x) e [3 C3x-1) [2 d.) C3x-1) - 3earrow_forward
- Suppose S = {r,u, d} is a set of linearly independent vectors. If x = 2r + 2u + 2d, determine whether T = {r,u, x} is a linearly independent set. %3| v Select an Answer 1. Is T linearly independent or dependent? Linearly Independent Linearly Dependent n-trivial linear relation below. Otherwise, enter O's for the coefficients.arrow_forward1. Consider the linear functionf: R3 R3 defined by f (x, y,z) = (x- 2y, 2x+ z, 3x- 2y +z). (a) Decide if the vector w = (-1,3, 2) is in R(f) and justify your answer. (b) Decide if the vector w = (0, 1, 2) is in R(f) and justify your answer. 2. Without doing any calculations, explain why the following sets of vectors are linearly dependent: (a) u= (2,-4) and v = (-6, 12) in R? (b) u= (1,0), v = (1, 1), and w = (2, -1) in R2 (c) u= (3,2, -1, -4), v = (1,0,0, 1), w = (0,0,0,0), and z (1,9,3,-1) in R T3(1,0.0), v = (0, 4, 0), w = (0, 0. –6), and = (1,5, –3) in R3 3. Determine if the following sets of vectors are linearly independent or linearly dependent. If they are linearly dependent, express one as a linear combination-of the others. (a) u= (3,5) and v = (15, 25) in R2 (b) u= (-1,-4) and v = (3, 12) in R2 (c) ū= (1,2,3), v = (3,2, 1), and w = (0,0, 1) in R (d) u=(-2,1,3), v= (2,9,-3), and w (2, 3,-4) in Rarrow_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
- Suppose S={r,u,d} is a set of linearly independent vectors. If x=3r+5u+3d, determine whether T={r,u,x} is a linearly independent set. 1. Is T linearly independent or dependent? If T is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients.arrow_forwardIf T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A= (see picture) and b= (see picture). Is the vector x found in the previous step unique?arrow_forward-3 [1 Let A 2 6 -4|,b : -9] 1 -7 T defined by T(X) = AX, find a vector X, whose 13 -5 image under T is b.arrow_forward
- Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,