4. (8 marks) Let V be the vector space of solutions to the ODE y" hyperbolic...
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...
Prove the following: (a) Let V be a vector space of dimension 3 and let {v,U2,U3} be a basis for V. Show that u2, u2 -2+s and uvi also form a basis for V (b) Show that1-,1-2,1-- 2 is a basis for P2[r], the set of all degree 2 or less polynomial functions. (c) Show that if A is invertible, then det A (Note: Show it for any det A-1 square matrix, showing it for a 2 x 2 matrix...
Let V be a finite-dimensional vector space, and let f :V + V be a linear map. Let also A be a matrix representation of f in some basis of V. As you know, any other matrix representation of f is similar to A. Show, conversely, that every matrix similar to A is a matrix representation of f with respect to some basis of V.
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
Let y be a differentiable function such that y' – 7y = 7, where y' is the first derivative of y. Define addition yı y2 = yı + y2 + 1 and scalar multiplication c Oyı = cyı + c – 1. Let V be the set of all differentiable functions that solve y' – 7y = 7. Note that these functions fit the form y = ket – 1. Determine if V is a vector space.
4. Let v={[a -.:a,nccc} Note that V is a vector space over R. View V as a R-vector space. (a) Find a basis for V over R. (b) Let W be the set of all matrices M in V such that M21 = -M12, where denotes complex conjugate. Show that W is a subspace of V over R and find a basis for Wover
I am having trouble with part B, i have done the other two parts. I know how to find the transition matrix when i have basis in the form of B = {(_ _ _), (_ _ _), (_ _ _)} but not one like this. Thanks so much for your help! Let V be the vector space of solutions to the ODE y" - 3y' +2y 0, with basis given by β-fet, e,2), and let z,-2et-e2t and z,-c"-et. (a)...
6. (a) Let V be a vector space over the scalars F, and let B = (01.62, ..., On) CV be a basis of V. For v € V, state the definition of the coordinate vector [v]s of v with respect to the basis B. [2 marks] (b) Let V = R$[x] = {ao + a11 + a222 + a3r | 20, 41, 42, 43 € R} the vector space of real polynomials of degree at most three. Write down...
5. The electric field in a certain region of space is given by the vector field Vector E(Vector r)= Vector E(x,y,z)= (x-z)hatx+(z-y)haty V/m Find any two points P(x1,y1,z1) and Q(x2,y2,z2) such that the electric field at P is perpendicular to the electric field at Q. Evaluate the electric field at each of these two points. (Hint: Use the dot product.).
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...