Question: Let V be a finite-dimensional vector space. Consider two linear operators T : V to V and S: V to V. (i) Show that the

Let V be a finite-dimensional vector space. Consider two linear operators T : V to V and S: V to V.

(i) Show that the product ST is invertible if and only if both S and T are invertible.

(ii) Prove that S T = idv if and only if T S = idv

(iii) Give an example showing that parts (i) and (ii) are false over an infinite-dimensional vector space.

Let V be a finite-dimensional vector space.

C13.l. Let V be a nite-dimensional vector space. Consider two linear operators T: V ) V and S: V > V. (i) Show that the product S T is invertible if and only if both S and T are invertible. (ii) Prove that ST 2 idv if and only if T S = idv. (iii) Give an example showing that parts (i) and (ii) are false over an innite-dimensional vector space

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

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

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Let V be a finite-dimensional vector space over F. The vector space V* = L(V,F) is called the dual of V. An element of V is thus a linear map f: V F, also known as a functional. (i) Let (,...,Un} be...

Let V be a finite-dimensional vector space and T, S : V V linear. Show that S = T-1 if and only if To S(x) = x for all x E V. Hint: show that (SoT)2 = SoT.

1. Given vector spaces U, V, W and linear transformations T : U - V and S : V - W we can define a new linear transformation R : U -> W such that R(u) = S(T(u)) for uEU. (a) Let V = C"(R) be the...

please solve all the parts 100% neat and clean thanks Question 4 Let V denote the real vector space of polynomials of degree at most 4. Define an inner product on V as follows: For polynomials f(x),...

We will consider the set X = Zn as a discretization of the circle. Let V be the real vector space of functions on X, i.e. V =R*. Let B = (@j , ..., en) be the basis of V defined by e ([k]) = 1 if [k]...

We will consider the set X = Z, as a discretization of the circle. Let V be the real vector space of functions on X, i.e. V = RX. Let B = (e,..., en) be the basis of V defined by e; ([k]) = 1 if [k]...

Need correct solution of this problem, asap Question 4 Let V denote the real vector space of polynomials of degree at most 4. Define an inner product on V as follows: For polynomials f (x). g(x) EV,...

P24.2. (i) Let T : V - V be a self-adjoint operator on a finite-dimensional inner product space. Assuming that 3 and 5 are the only eigenvalues of T, prove that 72 - 87 + 15 idy = 0. (ii) Exhibit...

Provide solutions.to the following attachments. Suppose that the business oyole in the United States is best described by R30 theory and that a new technology increases productivity. Show the effect...

Michael Spear invested $10,000 today in a fund that earns 8% compounded annually. Click here to view factor tables To what amount will the investment grow in 3 years? To what amount would the...

After completing your MBA, you are offered a job as a management consultant at a firm specialized in engagement and productivity. One of your first assignments is related to the education sector....

Jasmine Smith owns a condo worth $ 3 1 5 , 0 0 0 , a car valued at $ 2 9 , 0 0 0 , and miscellaneous assets worth $ 1 5 , 0 0 0 . She owes $ 1 8 5 , 0 0 0 on the condo and $ 1 2 , 0 0 0 on the car...

EXERCISES Q12.11 Baxter Inc. forecasts (in $ thousands) its future sales, NOPAT, and NOA for the next 4 years as follows: 2019A 2020E 2021E 2022E 2023E 7,500 8,300 9.100 10,000 7,000 900 1,100 1.350...