1. Show that the given vector functions are linearly independent on (-∞, ∞). X₁ (t) = = 0 X₂ (t) H 2t 3t² X3 (t) = 0 3+3
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- Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect. Please Provide Unique Answer. Thank you!Give an example of a linear combination that resulted in the zero vector that did NOT have 0's for the weights.8. (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a linearly dependent set in R*. (b) Express each vector as a linear combination of the other two.
- Let x(¹) (t) = -3t e 4e-3t, 0 x (²) (t) = [_5e-³]; x (³) (t) = -5e-3t, Are the vectors x(¹) (t), x(²) (t) and x(³) (t) linearly independent? choose ◆ If the vectors are independent, enter zero in every answer blank since those are only the values that make the equation below true. If they are dependent, find numbers, not all zero, that make the equation below true. You should be able to explain and justify your answer. 0 -3t [8] = 0[*]+[-+* 0 [4e-3t -5e-3t -0[ + -5e-3t -35e-3t -5e-3t -35e-3tExpress the vector = + y. F 10 as a linear combination of -[-] and ÿ==Show that the vectors X1 = 5e3t and (3)-* 2et X2 = Tet form a linearly independent set.
- 7. (a) Show that the vectors v1 = (0, 3, 1, – 1), v2 = (6, 0, 5, 1), and v3= (4, – 7, 1, 3) form a linearly dependent set in R4 (b) Express each vector as a linear combination of the other two.Find the value of 'Y' so that the three vectors are linearly dependent.2.30 Let r denote a position vector r = x = xiêį (r² = x₂x₁) and A be an arbitrary constant vector. Use index notation to show that: (a) (c) ² (r) = n(n+1) rn-2. V. (rx A) = 0. (b) (d) V(r. A) = A. ▼x (rx A) = -2A.