Question: Let L: Rn Rn be a linear operator defined by L(x) = Ax for x in R. Prove that if L is an isometry,

Let L: Rn → Rn be a linear operator defined by L(x) = Ax for x in R". Prove that if L is an isometry, then L-1 is an isometry?

Step by Step Solution

★★★★★

3.40 Rating (175 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock

Let Lx Ax It follows from the discussion preceding Theorem 79 that ... View full answer

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

Document Format (1 attachment)

Word file Icon

951-M-L-A-L-S (7166).docx

120 KBs Word File

Students Have Also Explored These Related Linear Algebra Questions!

Q:

Let L: V V be a linear operator defined by L(v) = cv, where c is a fixed constant. Prove that the representation of L with respect to any ordered basis for V is a scalar matrix.

Q:

Let T: C3 C3 be a linear operator defined by T(x) = Ax, where Find the kernel and nullity of T. =11 -i 1+i 0 1-i 1

Q:

Let L: M22 M22 be the linear operator defined by L(M) = MT. Find the matrix for L with respect to the standard basis for M22.

Q:

Let A bean n ( n orthogonal matrix, and let L: Rn - Rn be the linear operator associated with A; that is, L(x) = Ax for x in Rn. Let 9 be the angle between vectors x and y in R". Prove that the angle...

Q:

Let L: Rn Rm be a linear transformation defined by L(x) = Ax, for x in R". Prove that L is onto if and only if rank A = m.

Q:

1a) Let L : R n -> R m be a linear transformation. If Ker(L) = zero vector and Range(L) = R m , show that m = n. 1b) Let L : R n -> R m and M : R m -> R p be linear transformations. Show that Range(M...

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:

Consider the time-varying continuous-time system i (t) = A(t)x(t) + B(t)u(t), x(to) = To E Rn (1) where A : R - Rnxn, B : R - R"XP and u : R - RP are continuous functions. Also, denote the transition...

Q:

4. Let n E IN and fix p, q E R". Define the linear map L: R" -> R" by L(x) = (x . p)q, where we use the dot-product. You do not need to prove that L is indeed linear. What is the adjoint L* of L?...

Q:

9.5.6 Kernel Principal Component Analysis (KPCA) As discussed above, RKHS allows us to delinearize some linear algorithms. We give an example here with the Principal Component Analysis (see Exercise...

Q:

Suppose that the researchers who carried out the study described in the previous exercise wanted to estimate the mean reaction time with a margin of error of 5 msec. Using the sample standard...

Q:

In the appendix to this chapter, the balance sheet for the General Fund for the City of Atlanta is reported. 1. How is the format used on the balance sheet for the general fund different from the...

Q:

Steve Tague, Producing Artistic Director of the REP, said what, about how he is directing Medea. Group of answer choices People will not be bored. I am destroying the baggage. The production will be...

Q:

Which phases pose the highest amount of risk?

Q:

Graph the function. Be sure to label all the asymptotes. List the domain and the x-and y-intercepts? a. b. f(x) = (x-3) x + 3 f(x) = x -3x-4

Q:

Find a rational function that has vertical asymptotes x = -1 and x = 2 and x-intercept 1-4, 02?

Q:

The function S(t) = -16t2 + 64t + 192 gives the height S, in feet, of a model rocket launched with a velocity of 64 ft/sec from a hill that is 192 ft high. a) Determine how long it will take the...

Q:

Example Closed mortgage with prepayment privileges Home price = 125,000 Down payment 25,000 Conventional mortgage = 100,000 3 years term 8% per annum Amortization period of 25 years Closed mortgage...

Q:

Caspian Sea Drinks is considering the production of a diet drink. The expansion of the plant and the purchase of the equipment necessary to produce the diet drink will cost $25.00 million. The plant...

Q:

You are considering investing in a security that will pay you $5,000 in 26 years.a.If the appropriate discount rate is 9 percent, what is the present value of this investment?b.Assume these...