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- A natural cubic spline S is defined byS0(x) = 1+B(x −1)−D(x −1)3, if 1 ≤ x < 2 S1(x) = 1+b(x−2) −3/4(x−2)2 + d(x−2)3, if 2 ≤ x ≤ 3If S interpolates the data (1, 1), (2, 1), and (3, 0), find B, D, b, and d.For s=[−2,1,5] and t=[3,2,0] a) Determine s × t b) Show your work to confirm that s × t is orthogonal to both s and tA) compute x the component of x that is parallel to L and x the component of x that is orthogonal to L B) compute the reflection of x across the line L; refL(x)
- Q2. Ifw = (x^2 + y^2)xy, x = rcosθ, y = rsinθ a) Use chain rule to find∂w/∂r b) Find ∂w/∂rat the point r, θ = (2,π/4)a) Suppose T ∈ L(V ), m is a positive integer, and v ∈ V is such that Tm-1v does not equal 0 but Tmv=0. Prove that v, Tv, T2,........,T (m-1)v is linearly independent.10.Suppose that each of the vectors x(1), …, x(m) has n components, where n < m. Show that x(1), …, x(m) are linearly dependent. In each of Problems 11 and 12, determine whether the members of the given set of vectors are linearly independent for −∞ < t < ∞ . If they are linearly dependent, find the linear relation among them.
- Consider x1 = et and x2 = v(t)et.If x1 and x2 is linearly independent, then v′(t) = 0. Is this statement true or false?if u=f(x,y), x=g(w) and y=h(w), using chain rule, what is ∂u/∂w?express ∂w ∂u and ∂w ∂v using the chain rule and by expressing w directly in terms of u and v before differentiating. Then evaluate ∂w ∂u and ∂w ∂v at the point (u,v)= 1 2,−2.