Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, wedefine Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v E Vbe any non-zero vector.i. Prove that there is a positive ksn such that T k(v) = aoT o(v)+a1T 1(v)+ …+ak-1Tk-1(v) forsome ao,a1,...,ak-1EF.ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive naturalnumber n such that T k- (v) = aoT o(v) + aiT 1(v) + … + ak-iT kr-1(v) for some ao,a1,..,ak-1EF.Prove that a .= T 0(v),T 1(v),...,T kv-1(v) is linearly independent.

Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, we define Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v EV be any non-zero vector. i. Prove that there is a positive ksn such that T k(v) = aoT 0(v)+a1T 1(v)+ ·…+ak-1Tk-1(v) for some ao,a1,..,ak-1 EF. ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural number n such that T k- (v) = aoT o(v) + aiT 1(v) + ·… + ak-1T kr-1(v) for some ao,a1,..,ak-1 E F. Prove that a = T o(v),T 1(v),.,T kv-1(v) is linearly independent.

Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, we define Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v EV be any non-zero vector. i. Prove that there is a positive ksn such that T k(v) = aoT 0(v)+a1T 1(v)+ ·…+ak-1Tk-1(v) for some ao,a1,..,ak-1 EF. ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural number n such that T k- (v) = aoT o(v) + aiT 1(v) + ·… + ak-1T kr-1(v) for some ao,a1,..,ak-1 E F. Prove that a = T o(v),T 1(v),.,T kv-1(v) is linearly independent.

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

Related questions

Q: Let r1(t) and r2(t) be continuous vector functions with two components, and let t0 be a point in the…

A: Given- Let r1(t) and r2(t) be continuous vector functions with two components, and let t0 be a point…

Q: . Consider V1 1 and v3 = V2 = .0. 0 ) as vectors in R3. Let T: R3→R3 be a linear operator such that…

A: The solution is given as

Q: Determine the interval (s) on which the vector-valued function r(t) = ti + Vt + 1j + (t2 + 1)k is…

Q: Let f and g be any two smooth functions (defined on the same domain in R*), and let V = Vf and W =…

A: The given statement is

Q: 11. (a) Let u : R → R" and v : R → R". By writing u v as a summation, prove that (u · v)' = u' · v +…

A: Note: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question…

Q: ii) f x (g x h) is differentiable at t and d dt di {f x (g x h) } = (dg x h) +f x dt xh +fxg x dt dt

Q: Let V be the vector space of continuous functions and the scalar product in V is defined as (f,9)=[…

Q: Let V be an n-dimensional vector space. Define f: V→ R by f(v) = |v|". Let p,q E V.Find d fp(q).

Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

Q: et L be the linear operator on R? defined by L(x1, x2) = (x1 + 2x2,3x1)* Find the matrix A…

Q: et r be a three-dimensional real vector-valued function such that r′′ exists. Show that d/dt [r(t)…

A: Given: r is a real vector-valued function in three dimensions, and r'' exists. Our aim is to show…

Q: Let V be an n-dimensional vector space. Define f : V → R by f(V) = |V|2. Let p, q E V.Find d fp(q).…

A: Let V be an n-dimensional vector space. Define f:V→ℝ by fV=V2. If f is a smooth curve with regular…

Q: Create a vector of from F(x,y,z) such that the x, y, &z components contain at least two variables…

A: Given : vector of the form F(x , y ,z) To Find : vector of the form F(x , y ,z) such that the x, y…

Q: Let V be a vector space, and let T : V → M₂2 be linear. Let V₁, V₂ EV and suppose 2 T(v₁) = J 1…

Q: 2. Use abbreviated index notation to prove or disprove the following statements involving arbitrary…

A: Since there are multipart, so here find the solutions of first and second part.

Q: 2s Let H be the set of all vectors of the form Find a vector v in R' such that H span {i}. -s S

A: Spanning set of H

Q: Consider the space of square integrable functions L² ([–1, 1]). Let B = {Pn|n = 0, 1, 2, ...} the…

Q: I.) F = (y sin z)i + (x sin z)j + (xy cos z)k II.) F = e³+²z(i + xj + 2xk)

Q: Solve the differential equation by variation of parameters, subject to the initial conditions y(0) =…

Q: 1. Let P(R) denote the vector space of polynomials with coefficients in R. Define a function T: P(R)…

Q: Create a graph of the vector-valued function ř(t) = (t² – 1)i + (2t – 3)j, – 3 <t< 3. %3D

A: Introduction: A parametric equation is one in which the dependent variables are defined as…

Q: (6) Construct a non-negative finitely additive function lo on P (N) that is not a measure. Hint:…

A: Let S=ℕ and S=2S . For A∈S define μ(A)=0 if A is finite and μ(A)=∞ otherwise . For A1,A2.....An⊆S,…

Q: Let V be an n-dimensional vector space. Define f : V→R by f(v) =|v|². Let p, qe V. Find d fp(a).

Q: 21 Determine the domains of the vector-valued function r2(1) = (v/8 – 13, In 1, eviy %3D -

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

Q: defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2,…

Q: LA3-2. Suppose f is a nonzero linear function on some vector space V and U = ker f. Prove that V = U…

Q: iii) (f x g) xh is differentiable at t and d dt 7 {(f x g) x h} xgx h + x dt dg xh +(fx g) x dt dt

Q: 10. Let V be the vector space of functions spanned by B = {cos x – sin a, cos a + sin æ}. Find the…

Q: Let L: R2 → R2² be the linear operator defined by L 2

Q: Let F(R) be the vector space of all functions from R to R. Consider the functions v1, v2, v € F(R)…

A: Given vector functions, v1=sinx+x v2=e2x+2x v3=e2x-2sinx

Q: here

A: Here, we use Rank Nullity theorem. Definition of Jordan canonical form. Property of eigenvalue of…

Q: Let P2(R) be the vector space of polynomials over R up to degree 2. Consider {1+x – 2x?, –1 +x+x², 1…

A: Consider the provided question, Given that, B=1+x-2x2, -1+x+x2, 1-x+x2B'=1-3x+x2, 1-3x-2x2,…

Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let T : V → V be a linear operator on an n-dimensional F-vector space V. For any n E N+, we
define Tn.=T•T•…•T (n times) and To.=Iv,where Iv denotes the identity operator on V. Let v E V
be any non-zero vector.
i. Prove that there is a positive ksn such that T k(v) = aoT o(v)+a1T 1(v)+ …+ak-1Tk-1(v) for
some ao,a1,...,ak-1EF.
ii. Let kv denote the smallest number satisfying part (i), i.e. kv is the smallest positive natural
number n such that T k- (v) = aoT o(v) + aiT 1(v) + … + ak-iT kr-1(v) for some ao,a1,..,ak-1EF.
Prove that a .= T 0(v),T 1(v),...,T kv-1(v) is linearly independent.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Space$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

9. Let V be the vector space of all (2 × 2) matrices and define T: V → V by T(A) = AT. Show that I is invertible and give the formula for T−¹.
1. "Prove that the function T such that T(P(x)) = xP(x) + P'(x) is an operator of the vector space of polynomials (CC) cl
8. Show that an isometric linear operator T: HH satisfies T*T= I, where I is the identity operator on H.
a: Let T be a linear operator on a finite dimensional vector space V. Suppose T is diagonalizable. Simplify that V = Ker(T)® Im g (T). %3D
Let T: R² →→ R² where Find the matrix A such that T A = ([₁]) = ^ [*] · A T([7])=[52] and 1([2])-[1] -59 -43 T 469 238
6: Determine the domain and range of the vector valued function 1 r(t) = (In(t-2), t 4' fang (St
Q NO 3 Let T: P2(R) → P2(R) be a linear operator such that T(1) = 0, T(1 + x) = 1 and T(1+x²) = 2x. Explain whether T is diagonalizable or not? %3D
2. Let T: F→ F3 be the operator given by 4 3 3 T(v) = 4 3 1 4 1 3 Determine μT(x) and find a maximal vector for T. v.
Let V be the vector space of continuous functions and the scalar product in V is defined as (f,g)=L f(x)g(x)dx , then find the norm of vector f(x)=x+1 ? a. 2 b. 4 c. V3 d. 3