2. Let (X,(x) and (Y, (,)) be Hilbert spaces. For (x1,y1), (2, y2) = X x Y, put((x1,y1), (2, 2))xxy = (x1, x2)x + (y1, y2)Y.Show that (xxy is an inner product on the direct sum X x Y and it is a Hilbertspace under this inner product.

Answered: Image /qna-images/answer/2075d7d2-5674-4869-a2a8-16544cf38760.jpg

Answered: 2. Let (X,(x) and (Y, (,)) be Hilbert…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Consider the set C of complex numbers. Let (c², d) be a function where c² represents an element from the set of complex numbers and d(x, y) is defined as: - d(x, y) = √(L₁ – M₁)² + (L2 − M2)² where x = (L₁, L2₂) and y = (M₁, M₂), with L1, L2, M₁, M2 belonging to the set of complex numbers. Prove that (C², d) forms a metric space, where:
7. Prove that M2x2(C) is a complex vector space under the usual addition and scalar multiplication operations defined on 2 by 2 matrices.
b) For in R", we say that is in the kernel of Tif T(x) 0. If x₁ and 2 are both in the kernel of T, show that ax₁ + bx2 is also in the kernel of T for all scalars a and b. =
Q₂ Prove that such as ( x ₁ y ) + ( x ²₁ y ²) = (x + x ² + 1₂ y + y ² + ₁₂ and K(x, y) = (kxky) is not a vector space V = set of all pairs of real numbers. (x, y)
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22. Show that there do not exist scalars c₁, c₂, and c² such that c₁(1,0, 1,0) + c₂(1, 0, −2, 1) + c3(2, 0, 1, 2) = (1, -2, 2, 3)
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
Find the transpose, conjugate, and adjoint of (i) Transpose is idempotent: (4¹) = A. (ii) Transpose respects addition: (A + B) = A¹+ B¹. (iii) Transpose respects scalar multiplication: (c. A)¹ = c A. These three operations are defined even when m‡n. The transpose and adjoint are both functions from CX to Cxm These operations satisfy the following properties for all c E C and for all A, B € CX". (iv) Conjugate is idempotent: A = A. (v) Conjugate respects addition: A 6-3i 0 1 B = A + B (vi) Conjugate respects scalar multiplication: C. A = c. A. (vii) Adjoint is idempotent: (4†)t = 4. (viii) Adjoint respects addition: (A + B) =A+B+. (ix) Adjoint relates to scalar multiplication: (CA)t = c · At 2 + 12i 5+2.1i 2+5i -19i 17 3-4.5i

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2. Let (X,(x) and (Y, (,)) be Hilbert spaces. For (x1,y1), (2, y2) = X x Y, put ((x1,y1), (2, 2))xxy = (x1, x2)x + (y1, y2)Y. Show that (xxy is an inner product on the direct sum X x Y and it is a Hilbert space under this inner product.