4. Let X {(t,0): te R} U {[t, ∞o): te R}. Consider the binary relation Con X, that is, a C y if and only if x is a subset of y.(a) Show that is complete and transitive.(b) If u : X → R is a utility representation C, show that u((t,∞)) > u([t,∞))for every t€ R and u([t, ∞)) > u((s, ∞)) for every s, / ER with s < t.(c) Is there some utility u: X → R which represents ? Why or why not?

Given: X = {(t,∞) : t ∈ R} U [ t , ∞) : t ∈R }. Consider the binary relation ⊆ that is, x ⊂ y if and…

Answered: 4. Let X = {(t,co): te R} U {[t, oo):…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A E BL(H). Then A is called normal if A*A = AA*, uni- tary if A*A = I = AA*, that is, A* = A-¹, and self-adjoint if A* = A. Clearly, if A is unitary or self-adjoint, then A is normal. How- ever, a normal operator need be neither unitary nor self-adjoint. For example, let H = K² and for x = (x(1), x(2)) in H, A(x) = (x(1) − x(2), x(1) + x(2)). Then it can be easily seen that for x = H, A*(x) = (x(1) + x(2), −x(1) + x(2)), Request A*A(z) = 2(x(1),z(2)) = AA*(x). Hence A*A = AA*, but A*A# I and A* ‡ A. explain these steps in détail.
Suppose that T is a linear transformation from R" to Rm where m
If V = W1 e .. e Wk, then there exist k linear operators E,, .., E, On V such that (i) each E, is a projection (E = E;); (ii) E;E; = 0, if i# j; (iii) I = E1 + (iv) the range of E; is W; Conversely, if E, ..., E, are k linear operators on V which satisfy conditions (1), (ii), and (iii), and if we let W, be the range of E;, then V = W; e . OWg. + Eki ...
9. (Harder) show that given any three distinct points 2₁, 22, 23 in the z-plane, and any three points w₁, W2, W3 in the w-plane, there exists a Möbius transformation sending z; → w; for each j = 1, 2, 3. Wj
6. Find a basis for the null space of the functional f defined on R³ by ƒ(x) = §1 +§2 − §3, where x = (§₁, §2, §3).
Find 11 (3 (3x + 2y)dA where R is the parallelogram with vertices (0,0), (4,5), (-3,1), and (1,6). Use the transformation = 4u - 3v, y = 5u + v.
Show how a projective transformation in â: R°/~ → R°/ induces an affine map a if and only if â maps the line at infinity to itself. Prove that â induces a translation a if and only if â fixes each point on the line at infinity.
1) Let L: l2 → 12 be the left shift operator defined by I --. L(a1, a2, a3, ...) = (a2; a3, a4, ...) for all a = a) Show that L is a linear and continuous operator, that is L E CL(l2). Find ||L||, the (a1, a2, a3...) E l2- operator norm of L. b) Show that each point in the open disk D = {z €C: |z| < 1} is an eigenvalue of L. c) Show that o(L) = {z €C: |z| < 1} where o(L) is the spectrum of L. d) Show that op(L) = {z € C: |z| < 1} where op(L) is the point spectrum of L. %3|
2.) Prove that the identity transformation is linear; that is, prove that the function I:V→V defined by I (v)=v for every v in V is linear S
.Let X and Y be Banach spaces and let T/X -> Y be a continuous (bounded) linear transformation from Xb onto Y. Then T(A) is open set in Y for each open set A of X, i.e., T is an open mapping.
Show that T is a linear transformation

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