3. Let T: V→ V be a linear operator on a finite-dimensional F-vector space V. LetA₁,..., A E F be all the distinct eigenvalues of T. Show thatT is diagonalizableV = Ex₁ + ··· + Ex.(Since we proved in class that Ex₁,..., Ex are linearly independent, if V = Ex₁ ++ Ex then one can also write V = Ex₁ © · · · © Exk ·)Ө

Answered: Image /qna-images/answer/658d2f29-dea6-462a-a8f4-72c54fcefb45.jpg

Answered: 3. Let T: V→ V be a linear operator on…

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 be a vector space and let L: V → V be a linear transformation. Let A be a scalar. (a) Define VA, the A-eigenspace, and prove that it is a subspace of V. (b) Let v and v2 be eigenvectors with eigenval- ues A1 and A2, respectively. If A1 # 12, prove that v, and V2 are linearly independent.
3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0
(3) Let V be a finite-dimensional vector space and TE L(V). Let A1,...,Am denote the distinct nonzero eigenvalues of T. Prove that T is diagonalizable if and only if dim E(A1,T) + ...+ dim E(m, T) = dim range T.
5. Let v1, v2, ..., Vn E R". Let A be the n x n matrix whose j-th column is v; (j = 1, 2, n), that is, [v | А — V1 V2 V. Prove that V1, V2, ·.., Vn form a basis for R" det (A) + 0.
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.
9
7
Find a norm of the operator A(x) = (3 + 2t2)x(t) in space C[0, 1]. а. 3 4 Ob. 7 2 С. 3 O d. 6 е. 4
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. =
Theorem 3. Let A be a bounded linear operator acting between two real Hilbert spaces. Then 1. [R(A)] = Nr(A*). 2. R(A) = [N(A*)]*. 3. [R(A*)]+ = N(A). 4. R(A*) = [N(A)]+. Proof. Part 1 is just Theorem 1. To prove part 2, take the orthogonal complement of both sides of 1 obtaining [R(A)] = [N(A*)]*. Since R(A) is a subspace, the result follows. Parts 3 and 4 are obtained from 1 and 2 by use of the relation A** = A. I

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