Question: LetVbe a finite dimensional vector space andT:VVbe a linear transform. (i)Prove that if rank(T)= rank(T 2 ) (T 2 =TT, composition ofTwith itself ), then

LetVbe a finite dimensional vector space andT:VVbe a linear transform.

(i)Prove that if rank(T)= rank(T2) (T2=TT, composition ofTwith itself ), then range(T)

ker(T) ={0}.

(ii)Give an example to show that if rank(T)= rank(T2) does not hold, the conclusion in (i) may not be true.

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!

Q:

Explain the following que.stions as attached below. Pages 2-5 Q1 a) Let V be a finite dimensional vector space over a field F. If S and S' are two different bases of V, then prove that the number of...

Q:

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...

Q:

Microkernel operating systems aim to address perceived modularity and reliability issues in traditional "monolithic" operating systems. (i) Describe the typical architecture of a microkernel...

Q:

A discrete sequence {xn} can be converted into a continuous representation x(t) = ts X n= (t n ts) xn, where ts is the sampling period. (a) State two characteristic properties of Dirac's function. [2...

Q:

Developments in Technology Light is incident from air on the end face of a multimode optical fibre at angle of incidence as shown below. n n 1 2 The refractive indices of the core and cladding are...

Q:

Question -1 Consider the vector space of all 10x10 diagonal matrices. The dimension of this vector space is 20. select one: a. True b. False Question -2 If a vector space does not have a finite basis...

Q:

Indicate whether each statement is true or false. If true, prove the result. If false, provide a counterexample. (i) (2 points) There is a linearly independent subset of P2(R) consisting of four...

Q:

Please help Adobe Reader Touch X ASSIGNMENT 04 Due date: Thursday, 30 April 2020 UNIQUE ASSIGNMENT NUMBER: 897728 ONLY FOR SEMESTER 1 INSTRUCTIONS FOR THE ASSIGNMENT Take care to explain all your...

Q:

Can someone help me with these? I know these are pretty lengthy but it is midterm season and I cant figure these out/have the time for it TT. A function is bijective if it is both injective and...

Q:

Let V and W be finite dimensional vector spaces over F and T : V W is a linear map. Prove that T is an isomorphism if and only if {T(v 1 ),T(v 2 ),...,T(v n )} is a basis for W whenever {v 1 ,v 2...

Q:

Find the value of the integral s ( x A) da if the vector A = ai + zj + xk and S is the surface defined by the paraboloid z = 1 x2 y2, where x > 0.

Q:

Considering the regression ouput below, about what percentage of the variability is described by the model? Round to the nearest whole number. Standard T Parameter Estimate Error Statistic CONSTANT...

Q:

Question content area top Part 1 Which of the following would be considered a cash equivalent? Question content area bottom Part 1 A . Cashier ' s checks B . Money orders C . Certificate of deposit D...

Q:

2023 1040 form 1.Phillip and Claire are married and file a joint return. Phillip is self-employed as a real estate agent, and Claire is a flight attendant. Phillip and Claire have three dependent...