**Question 9:**Let \( T \) be an operator on an \( n \)-dimensional vector space \( V \) over a field \( F \).(a) If \( T \) has \( n \) distinct eigenvalues, prove that \( T \) is diagonalizable.(b) If \( T \) is diagonalizable, prove that the characteristic polynomial of \( T \) splits over \( F \).(c) Give an example of an operator \( T \) such that the characteristic polynomial of \( T \) splits but \( T \) is not diagonalizable.

Answered: Image /qna-images/answer/366fd1d0-e753-4a93-a836-5493cffac47e.jpg

Answered: 9. Let T be an operator on an…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 47CR: Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}

See similar textbooks

Similar questions

Consider the vector space V of all the real polynomials of degree less than or equal to three, p(x) = ax^3 + b x^2 + c x + d, and consider the linear transformation T: V ------> V given by the derivative, i.e., T( p ) = p' = 3ax^2 + 2 b x + c. Find the eigenvalues and eigenvectors of this transformation.
The action of a projection (generating) operator a basis function always creates a normalized linear combination with the symmetry properties of one of the irreducible representations. True False
Chetan
}) Let P3 be the vector space of all polynomials of degree 3 or less in the variable r. Let P1 (x) P2 (1) P3 (x) 2-z+ z? - r, 3z + 3z2-3r, 2+ z-z', - 2r3 6. PA (x) = 3-z+ 2z? and let C = {p1 (x), P2 (æ), P3 (2), P4 (x)}. choose a. Use coordinate representations with respect to the basis B {1, r, r-,r'} to determine whether the set C forms a basis for Pg. b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. { } c. The dimension of span(C) is S07 PM
Linear Algebra:Consider the polynomials p1(t)=1+t^2, p2(t)=t, p3(t)=t−t^2. Is {p1,p2,p3} a linearly independent set in P3?
Which of the listed sets is a vector subspace of the space V=R^2 over the field R?
(1) Let B = {√₁, 72, 73} be the basis of R³ defined by 1 *-(). *- (). *-(1) ), = = = Find the dual basis of B.
(b) Let f V -» V be any linear map of vector spaces over a field K. Recall that, for any polynomial p(X) = 0 X e K[X] and any v E V, p(X) vp(v) = ef*(v). i-0 The kernel of p(X) is defined to be Ker(p(X)) {v e V : p(X) v 0}. (b.1) Show that Ker(p(X)) is a linear subspace of V. When p(X) = X - A where E K, explain that Ker(P(X)) is the eigenspace of f with respect to A. (b.2) Let p(X) and q(X) be polynomials in K[X] so that gcd(p(X), q(X)) = 1. Show that Ker(p(X)q(X)) — Ker(p(X)) + Кer(4(х)) is a direct sum of subspaces. (Hint: Use the fact that if gcd(p(X), q(X)) = 1 then there exist a(X), Ь(X) € К|X] so that a(X)p(X) + b(X)q(X) %3D 1.) (b.3) Let c(X) e K[X] be any nonzero polynomial so that c(f) = 0 (for example c(X) is the characteristic polynomial of f). Suppose с (X) — р1 (X)р:(X)... Pm(X) where each p,(X) e K[X] and gcd(pg(X), p3(X))= 1 for all pairs 1 i
Let C2(-∞,∞)={f(x) in C(-∞,∞)|f''(x) exists for all x} be the set of differential functions. Show this is a subspace of C(-∞,∞).
4. (a) Let {b1, b2, b3, b4} be an orthonormal basis for a three-dimensional subspace of R6, and let u = b, - 2b2 + b3 - 2b4 and v= b, + b2 - b3 – bạ. Then = u'v . _? Show your work. (b) What is the length of u + v? Show your work.
Which of the following subsets of the vector space V=R^2×2 over the field R is not a vector subspace? (a) The set of all upper triangular matrices from V. (b) The set of all diagonal matrices from V. (c) The set of all matrices from V with zero determinant. (d) The set of all matrices from V with zero trace. (e) The set of all matrices from V having coefficient 0 at the intersection of the first row and the first column.
`

SEE MORE QUESTIONS

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS