6. Let |V(12)) be a state on H") 8 H2). Let {|k)} be an orthonormal basis for H") and{\j)} an orthonormal basis for H2). Show that|V(12)) = Ik) ® (k|V(12))and

Let |Ψ12> be a state on Η1⊗Η2. Let |k> be an orthonormal basis for Η1 and |j> an…

Answered: 6. Let (12)) be a state on H(¹) H(2).…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

4. Let V = sp({(1, 1, 1, 1)ª, (1, 1, 2, 4)¹, (0, −1, 6, 7)¹}) ≤ R¹. (a) Find an orthonormal basis {u₁, U2, u3} for V. (b) Write the vector (3,4, —2, −1)ª € V as a linear combination of u₁, U2, U3. (c) Find the vector v € V closest to x = = (4, 6, 1, 1)T.
a) Consider a function space in (-∞, ∞) with orthonormal basis {n}. Show that any function f(x) expanded in this basis as f(x) = En Cnon(x), has expansion coefficients given by = (onlf). Cn = b) Now consider another function g(x) expanded in the above basis as g(x) = Σn dnon (1). Prove that (flg) = Σncdn. c) Now consider a non-orthonormal basis {x} for this space, such that (XmIXn) = Smn. Gen- eralise the previous result to find an expression for (flg) in terms of Snm. Confirm that if Smn = 6mn, then this result reduces to the above one.
26. Prove: If {u1, u2, . . . , u,} is an orthonormal basis for R", and if A can be expressed as A = c,u,u + c,u,u + · · + c,,4,U, then A is symmetric and has eigenvalues c1, C2, . . . , Cn .
2 1 1 1 1 1 12, 2
3.) Apply the Gram-Schmidt orthonormalization process to the basis B = {1,t, t²} in P2 using the inner product 1 (p, q) = | p(t)q(t)dt -1 Show all calculations.
7
10. Suppose that |a) = E"-1 ax|k), |B) = E"-, bu|k). Show that (la)(8|)aj = a,b,, and that this means that the matrix representation in the given orthonormal basis satisfies |a)(3| = a b°, where a= [a a,]", b= [b, ..
6. Let A be an n x n hermitian matrix, i.e. A = A*. Assume that all n eigenvalues are different. Then the normalized eigenvectors { v; : j = 1,...,n} form an orthonormal basis in C". Consider B:= (Ax – ux, Ax – vx) = (Ax – ux)*(Ax – vx) | where (, ) denotes the scalar product in C" and µ,v are real constants with µ 0.
4. Let the vector space P([-1,1]) have the inner product = f(t)g(t)dt, using Gram-Schmidt procedure, convert the basis {1, x,x²} into orthonormal basis.
12. Let T be a linear operator on a finite-dimensional vector space V, and let A be an eigenvalue of T with corresponding eigenspace and generalized eigenspace Ex and KA, respectively. Let U be an invertible linear opera- tor on V that commutes with T (i.e., TU = UT). Prove that U(Ex) = Ex and U(K) K₁. =
12. Let B = {1+2x, x - x²,x + x²}. (a) Show that B is a basis for P2. (b) Let p(x) = 1+ 3x + x². Compute the coordinate [p(x)]g of p(x).
14. Consider the basis B = {1 +x + x², x + x², x²}. for P2. Let p(x) = 2 + 3x + 2x². Then [p(x)]B 2 2 O None 1 2

SEE MORE QUESTIONS

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS

6. Let (12)) be a state on H(¹) H(2). Let {[k)} be an orthonormal basis for H(¹) and {)} an orthonormal basis for H(2). Show that and |(¹2)) = |k) (k|(¹2)) k |(¹2)) = [(j|(¹2)) [j). j