Question: Let L be a linear operator on a vector space V. Define Ln, n 1, recursively by L1 = L Lk+ l(v) = L(Lk(V))

Let L be a linear operator on a vector space V. Define Ln, n ≥ 1, recursively by
L1 = L
Lk+ l(v) = L(Lk(V)) for all v ∈ V
Show that L" is a linear operator on V for each n > 1.

Step by Step Solution

★★★★★

3.23 Rating (158 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

The proof is by induction on n In the case n 1 L 1 is a linear operator sinc... View full answer

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Document Format (1 attachment)

949-M-L-A-E (624).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Let L be a linear operator on R1 and let a = L(l). Show that L(x) = ax for all .v R1

Let L be a linear operator on Rn. Suppose that L(x) =0 for some x 0. Let A be the matrix representing L with respect to the standard basis [e1, e2,..., en]. Show that A is singular

Let L: V V be a linear operator. A nonempty sub-space U of V is called invariant under L if L(U) is contained in U. Let L be a linear operator with invariant subspace U. Show that if dim U = m and...

Let L be a linear operator on a vector space V. Let A be the matrix representing L with respect to the ordered basis [v1,...,vn] of V [i.e., L(j) = Show that A'" is the matrix representing L'" with...

8.1 WRITE on paper for upvote. Section: 25, 32, 33 please 8.1 General Linear Transformations 20. Consider the basis S = {V1, V2} for R', where v, = (-2, 1) and 9. T : Rx - R., where V2 = (1, 3), and...

Math 636 Final - Quiz Component 1. The system of linear equations 2x2 x3 = 3 2x1 + x2 + 2x3 = 1 x1 + x2 + 3x3 = 2 (a) is consistent with a unique solution. (b) is consistent with infinitely many...

(a) Find the transition matrix S from B' to B. (b) Use the matrices S and A to find [v], and [T(v)], where [V]B/ = (-1,4) T. (c) Find S" and A' (the matrix for T relative to B' ). (d) Find [T(v)]B,...

6. Let e = (v1, ..., vn) be an ordered basis of an n-dimensional vector space V , and let Le : V K n be the coordinate map given by Le(v) = (a1, a2, ..., an) K n , where the ai are such that v =...

4 This assignment, based on the content of Unit 4 of the Study Guide, is worth 3% of your grade. We recommend that you hand it in after you complete Unit 4. You must show all of your work in order to...

ANSI-SPARC6 Programming Language Compilation Write notes on each of the following topics: (a) the implementation of labels and jumps in a recursive, block structured programming language [7 marks]...

The table shows the numbers of acres planted, the numbers of acres harvested, and the annual yields (in pounds) of spinach for five years. Use technology to find (a) The multiple regression equation...

Determine the order of the following events involving lipoprotein-mediated transport of ______ dietary triacylglycerols and cholesterol. ______ Triacylglycerols are removed from circulating VLDL by...

2. What are the two objectives of accounting for income taxes?

2. Suppose that the United States and Mexico are the only two countries in the world and that labor is the only productive input. In the United States, a worker can produce 12 bushels of wheat or 2...

Find A when (a) (b) (c) (d) T 1-1 (2.AY=| 2 3 (221 1 2 2 1 2 (4-1-21)' =-21 1.0

Find A when: A"l=112 1

(a) In the system 3x + 4y = 7 4v + 5 y = 1 substitute the new variables xÊ¹ and yÊ¹ given by x = - 5xÊ¹ + 4yÊ¹ y = 4xÊ¹ - 3yÊ¹. Then...

You expect to receive a payout from a trust fund in 3 years. The payout will be for $13,400. You plan to invest the money at an annual rate of 5.3 percent until the account is worth $22,600. How many...

New Bank has total checkable deposits of $90 million in its account. Its capital, including common stocks, retained earnings, preferred stocks and subordinated debt is $5 million. The reserves...

Get Annual Reports for Home Depot and Lowes from web for 2014 and 2015 (they may not end in the same month and thats fine). These reports will have voluminous data but you only want to focus on...