I use definition of basis to solve this problem
Let W denote the set of smooth functions f(x) in CⓇ such that f"(x) = -f(x)....
(5) Let W denote the set of smooth functions f(2) in CⓇ such that f'(x) = -f(L). That is, W= {f() in C | F"(x) = -f(x)} In the previous worksheet, we showed that: • W is a subspace of Cº. . For all a and b, a sin(2) + b cos(x) is in W. (a) Show that (sin(x), cos(x)} are linearly independent. Hint: Set an arbitrary linear combination equal to 0, and show the coefficients must be 0. (b)...
Let Coo denote the set of smooth functions, ie, functions f : R → R whose nth derivative exists, for all n. Recall that this is a vector space, where "vectors" of Coo are function:s like f(t) = sin(t) or f(t) = te, or polynomials like f(t)-t2-2, or constant functions like f(t) = 5, and more The set of smooth functions f (t) which satisfy the differential equation f"(t) +2f (t) -0 for all t, is the same as the...
Problem 4. Let V be the vector space of all infinitely differentiable functions f: [0, ] -» R, equipped with the inner product f(t)g(t)d (f,g) = (a) Let UC V be the subspace spanned by B = (sinr, cos x, 1) (you may assume without proof that B is linearly independent, and hence a basis for U). Find the B-matrix [D]93 of the "derivative linear transformation" D : U -> U given by D(f) = f'. (b) Let WC V...
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
We say that an nxn matrix is skew-symmetric if A^T=-A. Let W be the set of all 2x2 skew-symmetric matrices: W = {A in m2x2(R) l A^T=-A}. (a) Show that W is a subspace of M2x2(R) (b) Find a basis for W and determine dim(W). (c) Suppose T: M2x2(R) is a linear transformation given by T(A)=A^T +A. Is T injective? Is T surjective? Why or why not? You do not need to verify that T is linear. 3. (17 points)...
Question 5 Is the set of functions linearly dependent or linearly independent? f(x) = 7, g(x) = 5x +1, h(x) = 3x2 - 4x + 5 Linearly dependent Linearly independent Have no clue... Question 6 Given a solution to the DE below, find a second solution by using reduction of order. r’y' – 3xy + 5y = 0; y1 = r* cos(In x) y2 = xsin(In x) y2 = x2 sin Y2 = 2 * sin(In) . . y2 =...
Suppose f(x,y) is such that V f is continuous everywhere. Let C be the smooth curve given by F(t) = (cos(t), cos(t) sin(t)) for 0 <t< 7/4. Suppose we know that f(0, 1) = 3, $(1,0) = 7, f (VE) = 2, 2' 2 Use this information to find Sc Vf. dr. Show all work and expain your reasoning.
2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a E R3 with R3 be smooth with = 1 and curvature k and torsion r, both Assume there exists a unit Ta constant = COS a. circular helix is an example of such curve a) Show that b) Show that N -a 0. c) Show that k/T =constant ttan a 2. Let (a, b) nonvanishing. Denote the Frenet frame by {T, N, B} vector a...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
2. Derivation of the Kronig-Penney result: Write the solution for the wave function in the region 0 < r < a as a linear combi nation of sin and cos functions and in the region-b< 0 as a linear combination of the hvperbolic sin and cos functions: ψ(z) Asin Kr +Bcos Kx (0<x <a) = By invoking the continuity and differentiability conditions at a- 0, show that B D and AK - CQ, so essentially two constants, say A and...