a. Let be an differential operator. Show that L is a linear operator.
b. Let be an differential operator. Show that the kernel of L is a vector space
c. Let . Show that the set of functions which satisfy L(u) = g(x,t) form an affine linear subspace.
a. Let be an differential operator. Show that L is a linear operator. b. Let be an differential operator. Show that...
(3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t) (3) Let L (a20- 0). Show that the set of functions which satisfy L(u) form an affine lincar subspace. = g(x,t)
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W (6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
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...
4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W be the nll space of T - c/. (a) Prove that W is the subspace spanned by 4 (b) Find the monic generators of the ideals S(u;W), S(q;W), s(G;W), 1 4. Let T be the linear operator on F which is represented in the standard ordered basis by the matrix c0 0 01 Let W...
3.[4p] (a) In the following questions assume that a linear operator acts from a finite- dimensional linear space X to X, and assume that the word "vector means an element of X. Recall that a vector a is a pre-image of a vector y (and y is the image of x) for a linear operator A: X -> X, if Ax-y. How many of the following statements are true? (i) A linear operator maps a basis into a basis. (ii)...
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that we have shown in class that in this case, there exists a unique unitary operator U such that A=UA. The point of this exercise is to prove the following result: an invertible operator A is normal if and only if U|A= |AU. a) Show that if UA = |A|U, then AA* = A*A. Now, we want to show the other direction, i.e. if AA*...
Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned by γ. Define γ′ to be the ordered set obtained from γ by reversing the...
(1 point) Let L be the linear operator in R? defined by L(x) = (4x1 – 2x2, -6x1 + 3x2) Find bases of the kernel and image of L. 00 Kernel: * Image: [-2,3] To enter a basis into WebWork, place the entries of each vector inside of brackets, and enter a list of these vectors, separated by commas. For instance, if your basis is 1 2,1l/, then you would enter [1,2,3], 31 [1,1,1) into the answer blank.
Problem 5. A subset A c Rn is an affine subspace of Rn if there exists a vector b є R', and a underlying vector subspace W of Rn such that (a) Describe all the affine subspaces of R2 which are not vector subspaces of R2. (b) Consider A є Rnxn, b є Rn and the system of linear equations Ax-b Prove that (i) if Ar= b is consistent, then its solution set is an affine subspace of Rn with...