Question: Prove that x R n solves the linear system A x = b if and only if x T A T v = b

Prove that x ∈ Rⁿ solves the linear system A x = b if and only if x^TA^Tv = b^Tv for all v ∈ R^m. The latter is known as the weak formulation of the linear system, and its generalizations are of great importance in the study of differential equations and numerical analysis, [61].

Step by Step Solution

★★★★★

3.51 Rating (161 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

To prove that x Rn solves the linear system Ax b if and only if xT AT v bTv for all v Rm we need to ... 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

Students Have Also Explored These Related Applied Linear Algebra Questions!

If A and B are sets, show that A B if and only if A B = A.

If a > 0; b > 0; and n N, show that a

The following provides an alternative way to establish Lagrange's Theorem. Let G be a group of order n, and let H be a subgroup of G of order m. (a) Define the relation R on G as follows: If a, b G,...

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...

Let r and s be solutions to the quadratic equation x 2 b x + c = 0. For n N, define d0 = 0 d1 = r s dn = b dn1 c dn2 (n 2) Prove that dn = r n s n for all n N. [4 marks] (b) Recall that a commutative...

A) An adjacency matrix has as many rows and columns as there are nodes. The entries in an adjacency matrix are 0's and 1's. If there is an edge from node N to node M (in other words, if N and M are...

12.19 LAB: Solution of linear systems of equations Solution of systems of multiple equations in multiple unknowns is a common engineering task. When those equations are linear, meaning the unknowns...

Answer the following. Example 1: A C B Given: AD = BE ; DC = EC ; C bisects AB. Prove: AADC = ABEC D E Proof: Statements Reasons 1 . 1. 2. 2. 3. 3. Example 2: C Given: 21= /2; ZA= ZE ; N is the...

10.2 (a) Write a wl-commented MaTLAB function program mysolvecheck with input a number n that makes a random n n matrix A and a random vector b, solves the linear system Ax-b, calculates the scalar...

Q1: Write a well-commented Matlab function program mygrowth that gives the population for the next 100 years. The birthrate should be 0.05, and the initial population, p0=1000. Q2)Modify mygrowth...

Vons Grocery, a large retail grocery chain in Los Angeles, sought to acquire Shopping Bag Food Stores, a direct competitor. At the time of the proposed merger, Vons sales ranked third in the Los...

The formula to find the surface area of a cylinder is SA =2 279 5.) The formula to find the surface area of a cylinder is SA=2rr+ 2nrh. If the diameter of the base is 15 m and the cylinder stands 12...

answer this Fill in the Blank Question [) ) capital is the amount of money paid into a company by its owners

Referring to Exercise 14-50, another student in the same class computed a regression equation relating the same two variables. The slope of the equation was found to be -0.735. After trying several...

Let w(x) > 0 for a (a) Prove that defines a norm on C°[«. h ]. called the weighted L1 norm. (b) Do the same for the weighted L norm ||f||.u, = max{|f(x)| w(x) ; a ¤ x ¤ h }....

Let || ||1 and || ||2 be two different norms on a vector space V. (a) Prove that ||v|| = max{ ||v||1. ||v||2} defines a norm on V. (b) Does ||v|| = min{ ||v||1. ||v||2} define a norm? (c) Does the...

Answer Exercise 3.3.1 for a. b. c. 2 04

Imperial Jewelers manufactures and sells a gold bracelet for $406.00. The company's accounting system says that the unit product cost for this bracelet is $268.00 as shown below: Direct materials...

Please explain how you got the answer. 38. The entry to record the transaction will consist of a debit to Cash for $1,800,000 and a credit or credits to A) Preferred Stock for $1,800,000. B)...

A company has an opening debit balance in Allowance for Doubtful Accounts of $ 5 0 0 . Its sales for the year total $ 5 0 0 , 0 0 0 . Management estimates that 1 % of sales are uncollectible. What is...