(Part b) Write the vector sum F1+F2 +F3 in terms of the unit vectors i and j. Use Fi = Fi), F, = |F2), and F3 = |F3| to be the magnitude of the vectors Fi, F2, and F3, respectively. Drag the appropriate terms to construct the correct expression. Pay attention to the difference between a and in the trigonometric functions. Ē + F2 + Fz (F1 + F2 + F3 + + F2 + F3 · α E +X...
3. (10 points) Let F denote the vector space of functions f: R + R over the field R. Consider the functions fi, f2. f3 E F given by f1(x) = 24/3, f2() = 2x In(9), f() = 37*+42 Determine whether {f1, f2, f3} is linearly dependent or linearly independent, and provide a proof of your answer.
(1 point) Calculate the Wronskian for the following set of functions: f1(x) = 0, f2(2) = 2.c +5, f3(2) = 1e" + b W(fi(2), f2(2), f3()) NO_ANSWER 1. Is the above set of functions linearly independent or dependent?
5. Let V be the subset of Cº(R) consisting of all functions that can be expressed in the form a sin 2x + be 4x + cos2r for a, b, c ER. (a) (4 points) Prove that V is a subspace of C(R). (b) (3 points) Let fix) = sin 2x + e4r f2(x) = sin 2.c + cos 2.0 f3(x) = 4x + cos2r. The set B = (f1, f2, f3) is an ordered basis for V. (You do...
Consider the linear transformation T from V = P2 to W = P2 given by Tlao + ayt + azt?) = (-620 + 3a1 + 2a2) + (200 + 204 + 2az)t + (420 + 3a1 + 4a2)t? Let F = (f1, f2, f3) be the ordered basis in P2 given by fi(t) = 1, f2(t) = 1 + t, fz(t) = 1 + + + + Find the coordinate matrix [T]FF of T relative to the ordered basis F...
11 1 11 3. Let f1, f2, ..., fn are differentiable functions from V HR with V a Hilbert space. Define F :V HRN by F(a) = [fi(x)] f2(2x) I:T, V x EV, [fn (2)] where R” is endowed with the standard inner product. Prove that T(V fi(2), y) | |(Vf2(2), y) | DF(x)(y) = V rEV. Liv fn(x), y)]
Determine whether the given set of functions is linearly independent on the interval (−∞, ∞) f1(x) = x f2(x) = sin(x) f3(x) = sin(2x)
Let fi and f2 be functions such that lim e s f1 (2) = + and such that the limit L2 = lim a s f2 (x) exists. Which one of the following is NOT correct? O limas (f1f2)(x) = 0 if L2 = 0. limas (fi + f2)(x) = too if L2 = -0. Olim as (f1f2) (x) = too if 0 <L2 5+co. lim a s (f1f2)(x) = - it L2 = -. Which one of the following...
Let X = ℝ with the standard topology and I = [0, 1]. Let F1 be the subset of I formed by removing the open middle third (1/3, 2/3). Then F1 = [0, 1/3]⋃[2/3, 1] Next, let F2 be the subset of F1 formed by removing the open middle thirds (1/9, 2/9) and (7/9, 8/9) of the two components of F1. Then F2 = [0, 1/9] ⋃[2/9, 1/3] ⋃[2/3, 7/9] ⋃[8/9, 1] Continuing this manner, let Fn+1be the subset of...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...