In Problems 13-19, determine whether the given
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Fundamentals of Differential Equations and Boundary Value Problems
- Let L: R2 → R2 be a linear operator. If L ((1, 2)T ) = (−2, 3)T and L ((1,−1)T ) = (5, 2)T find the value of L ((7, 5)T ).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_forwardProblem 4. Show that the following functions are linearly independent in the vector space C[0, 1]: (i) e, e- and e²x; (ii) 1, e+e- and e- e-.arrow_forward
- 3. Suppose A = 3 1 1 -2 (a) Find a vector X E R² such that AX = (b) Find a vector X € R² such that AX = B 9arrow_forward* 4: Which of the following functions has gradient vector [4x₁x₂ 2x² 3x3]¹? A: f(x) = 2x³ + 2x3 + x²x² B: f(x) = 2x²x₂ + x³ C: f(x) = 2x²x₁ + x² D: Non of these A B Carrow_forwardSuppose 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_forward
- Problem 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_forwardEvaluate the vector-valued function at each given value of t. r(t) = (2t + 1)i + t2j − √t + 2k (a) r(0) (b) r(−2) (c) r(c − 1) (d) r(1 + ∆t) − r(1)arrow_forwardx 2y 3. Let T ((1)) = (1 + 2) 4x a) T can also be written as T b. matrix you found above. (Ⓒ)) = ¹ (*) A c) Use your answer to b) to find a vector (Ⓒ) for some matrix A. What is the matrix A? ·((*)). such that T || = d) Use your answer to b) to find a solution to the system of equations x + 2y = = 2 4x + 9y = -1arrow_forward
- (a) Is in ker L? (b) Is in ker L? (c) Is in range L? (d) Is in range L? (e) Find ker L. (f) Find a set of vectors spanning range L. 2. Let L: R2 → R? be the linear operator defined byarrow_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_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
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