Question: Let L: U V be a linear function between inner product spaces. Prove that u R n solves the inhomogeneous linear system L[u]

Let L: U → V be a linear function between inner product spaces. Prove that u ∈ Rⁿsolves the inhomogeneous linear system L[u] = f if and only if Explain why Exercise 3.1.11 is a special case of this result. Remark. Equation (7.83) is known as the weak formulation of the linear system. It plays an essential role in the analysis of differential equations and their numerical approximations, [61].

Data From Exercise 3.1.11

Prove that x ∈ Rⁿ solves the linear system Ax = 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].

(u, L* [v]) = (f,v) for all VEV. (7.83)

Step by Step Solution

★★★★★

3.44 Rating (160 Votes )

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock

To prove that u Rn solves the inhomogeneous linear system Lu f if and only if u Lv f v for all v V l... 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!

Let f L(X, Y) be a linear function between Hilbert spaces X and Y. 1. There exists a unique x* X such that fy(x) = xT x*. 2. Define f*: Y X by f*(y) = x*. Then f * satisfies f (x)T y = xT f*(y) 3....

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 w1,... wn be any basis of the subspace W Rm. Let A = (w1,...,wn) be the m x n matrix whose columns are the basis vectors, so that W = rng A and rank A = n. Let P = A(AT A)-1 AT be the...

How would you modify the statement of Theorem 7.62 if ker L {0}? Theorem 7.62 Suppose L: U V is a linear map between inner product spaces with ker L = {0} and adjoint map L*: V U. Let K = L* L: U...

COMPLETE SLUTION IN WRITTEN FORM NO IMAGEAND ONLY ORIGINAL WORK \fAssume that f(x) = 6x + 14 and g(x) = 5x - 1. Find the composition (fog)(x) (A) 30x + 20 (B) 30x + 13 (C) 30x + 69 (D) 30x + 8 The...

1 1. (6+4+4+6) Let P3 denote the vector space of all third degree or less polynomials. (a) Find two linearly independent vectors in P3 that interpolate the data points (1, 2), (2, 1) and (3, 2). (b)...

Math 400 Assignment #3 -- Numerical Integration and Differentiation 1. For the function g x e x 2 2 Due: 6am April 09, 2016 use the centered difference formula with h 101 , 102 , , 1020 to create a...

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...

MATHEMATICS FOR MACHINE LEARNING Marc Peter Deisenroth A. Aldo Faisal Cheng Soon Ong Contents Foreword 1 Part I Mathematical Foundations 9 1 Introduction and Motivation 11 1.1 Finding Words for...

ESE 500: Homework 3 Due on October 16 (Monday) Part A: Matrix Exponential Problem 1. Without using Matlab, compute eAt , using two of your favourite methods, for: \u0014 \u0015 0 2 A= 1 2 It is...

Having a support team around you can be invaluable as a foundation for your leadership. When you are most in need of finding direction, your support team helps you get on track. Members of your...

For the Bigelow Manufacturing example in this chapter, re-create the simulation shown in Table 14.10 and Exhibit 14.7 using Crystal Ball. Assume that the repair time is normally distributed with a...

This year Diane intends to file a married filing jointly return. Diane received $ 2 0 1 , 7 0 0 of salary and paid $ 6 , 4 5 0 of interest on loans used to pay qualified tuition costs for her...

6. BY dynamic programming it is possible to obtain the best and next-best solution to a problem. Mark the statements as T (True) or F (False).

Suppose that hunters are allowed to shoot a fixed number of the Northern Minnesota deer in Exercise 8.1.6 each year. (a) Explain why the population model takes the form du/dt = .27 u - b, where b is...

(a) Prove that if ui (t) and u2 (t) are any two distinct solutions to du/dt = au with a > 0, then |u1(t) - u2(t)| as t . (b) If a = .02 and u1(0) = .1, u2(0) = .05, how long do you have to wait...

(a) Write down the exact solution to the initial value problem du/dt = 2/7u, u(0) = 1/3. (b) Suppose you make the approximation u(0) = .3333. At what point does your solution differ from the true...

If the going market rate of interest is 4%, how much would you have to invest each and every year to have $1,000,000 at the end of 20 years? If the going market rate of interest is 12%, how much...

Reagan Corporation is a wholesale distributor of truck replacement parts. Initial amounts taken from Reagan's records are as follows: Inventory at December 31 (based on a physical count of goods in...

Use the following to answer questions 20-23: On April 1, year 1, Beta Corporation issues $20 million of 12%, 10-year bonds payable at par. Interest on the bonds is payable semiannually each April 1...