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mathematics
linear algebra

Questions and Answers of Linear Algebra

Prove that (11.154) defines an inner product.
(a) Prove the addition formula (11.151) for the hyperbolic sine function. (b) Find a corresponding addition formula for the hyperbolic cosine.
Use the Green's function (11.152) to solve the Sturm-Liouville boundary value problem (11.145) when the forcing function is
(a) Find the Green's function for the mixed boundary value problem -u" + a)w2u = u(0), (0) = 0, u'(1) = 0.(b) Use your Green's function to find the solution when
Answer Exercise 11.5.4 for the Neumann boundary conditions t/'(0) = «'(1) = 0. Explain why, in contrast to the boundary value problem for a bar, this problem does have a Green's function and a
Explain why the homogeneous Neumann boundary value problem for a Sturm-Liouville operator with q(x) > 0 is positive definite. What is the minimization principle?
(a) For which values of λ. does the boundary value problem u" + λu = h(x), u(0) = 0, w(l) = 0 have a unique solution?(b) Construct the Green's function for all such k.(c) In the non-unique cases,
Show that each summand in the differential equation for a Sturm-Liouville boundary value problem is self-adjoint. Then use the method of Exercise 7.5.21 to recover the self-adjoint form (11.155).
(a) Show that any regular second order linear ordinary differential equationa(x)u" + b(x)u' -I- c(x)u = f(x),with a (x) ≠ 0, can be placed in Sturm- Liouville form (11.143) by multiplying
Use the finite element method to approximate the solution to the boundary value problemCarefully explain how you are setting up the problem. Plot the resulting solutions and compare your answer with
Consider the boundary value problem solved in Example 11.15. Let W" be the subspace consisting of all polynomials u(x) of degree < n satisfying the boundary conditions «(0) = n(l) = 0. In this
Consider the boundary value problem - u"+ λu = x, for 0 < x < π, with u(0) = 0, u(1) =0.(a) For what values of a does the system have a unique solution?(b) For which values of λ can you find
Given data points (xo, uo). .........(xn un) with x, ‰ , i ‰ j, prove that there is a unique continuous piecewise affine interpolant u = f(x), so u = 0..........n.(a) Prove that the
Prove that every continuous piecewise affine function based on the mesh points x0where(a) What are the values of a and b?(b) Write the hat function (11.168) in the form (11.179) .(c) Write the
For each of the following boundary value problems,(i) Solve the problem exactly.(ii) Approximate the solution using the finite element method based on 10 equally spaced nodes.(iii) Compare the graphs
The exact solution to the boundary value problem- u" = 3x, u(0) = u(1) = 0, is u(x) = u(x)- 1/2x-1/2x3.(a) Use the finite element method based on 5 equally spaced nodes to approximate the
(a) Devise a finite element method for solving themixed boundary value problem(b) Apply your method to approximate the solution to using 10 equally spaced mesh points. Compare your numerical
Consider the periodic boundary value problem - u" + u = x, u(0) = u(2π), u'(0) = u(2π).(a) Write down the analytic solution.(b) Write down a minimization principle.(c) Divide the interval [0, 2tc]
Answer Exercise 11.6.5 when the finite element subspace W" consists of all periodic piecewise affine functions of period 1, so φ(x + 1) = φ(x). Which version is better?
Use the method of Exercise 11.6.6 to approximate the solution to the periodic Mathieu boundary value problem -u" + (1 + cosx)u = 1, m(0) = u(2n), w'(0) = u'(2π). In this case, the exact solution
Give a direct proof that the finite element matrix (11.171) is positive definite, by either(a) Determining its L D LT decomposition, or(b) Finding its eigenvalues, using Exercise 8.2.48.
(a) Is the finite element matrix (11.171) diagonally dominant?(b) Does the Jacobi iteration scheme converge to the solution to Me = b? How fast? Hint: Use Exercise 8.2.48.
In each case verify that the following are solutions for all values of s and t. (a) x = 19t - 35 y = 25 - 13t z = t is a solution of 2x + 3y + 2 = 5 5x + 7y - 4z = 0 (b) x1 = 2s + 12t + 13 x2 =
Find the solution of each of the following systems of linear equations using augmented matrices. (a) x + y + 2z = -1 2x + y + 3z = 0 -2y + z = 2 (b) 2x + y + z = -1 x + 2y + z = 0 3x- 2z - 5
Find all solutions (if any) of the following systems of linear equations. (a) 3x - 2y = 5 - 12x + 8y = -20 (b) 3x - 2y = 5 - 12x + 8y = 16
In each case either show that the statement is true, or give an example3 showing it is false. (a) If a linear system has n variables and m equations, then the augmented matrix has n rows. (b) A
Solve the system 3x + 2y = 5 7x + 5y = 1 by changing variables x = 5xʹ - 2yʹ y = -7xʹ + 3yʹ resulting equation for xʹ and yʹ.
Find a, b, and c such that
Workmen John and Joe earn a total of $24.60 when John works 2 hours and Joe works 3 hours. If John works 3 hours and Joe works 2 hours, they get $23.90. Find their hourly rates.
Find all solutions to the following in parametric form in two ways. (a) 3x + y = 2 (b) 2x + 3y = 1 (c) 3x - y + 2z = 5 (d) x - 2y + 5z = 1
Write the augmented matrix for each of the following systems of linear equations. (a) x - ly = 5 2x + y = 1 (b) x + 2y = 0 y = 1 (c) x - y + z = 2 x - z = l y + 2x = 0 (d) x + y = 1 y + z = 0 z -
Write a system of linear equations that has each of the following augmented matrices.(a)(b)
Find the solution of each of the following systems of linear equations using augmented matrices. (a) x - 3y = 1 2x - 7y = 3 (b) x + 2y =1 3x + 4y = -1 (c) 2x + 3y = -1 3x + 4y = 2 (d) 3x + 4y = 1 4x
Which of the following matrices are in reduced row-echelon form? Which are in row-echelon form?(a)(b) (c) (d) (e) (f)
Find the rank of each of the matrices in Exercise 1.Exercise 1Which of the following matrices are in reduced row-echelon form? Which are in row-echelon form?(a)(b) (c) (d) (e) (f)
Find the rank of each of the following matrices.(a)(b) (c) (d) (e) (f)
Consider a system of linear equations with augmented matrix A and coefficient matrix C. In each case either prove the statement or give an example showing that it is false. (a) If there is more than
In each case, show that the reduced row-echelon form is as given.(a)(b)
Find the circle x2 + y2 + ax + by + c = 0 passing through the following points. (a) (-2, 1), (5, 0), and (4, 1) (b) (1, 1), (5,-3), and (-3,-3)
A school has three clubs and each student is required to belong to exactly one club. One year the students switched club membership as follows: Club A. 4/10 remain in A, 1/10 switch to B, 5/10 switch
Carry each of the following matrices to reduced row-echelon form(a)(b)
The augmented matrix of a system of linear equations has been carried to the following by row operations. In each case solve the system.(a)(b) (c) (d)
Find all solutions (if any) to each of the following systems of linear equations. (a) 3x - y = 0 2x - 3y = 1 (b) 3x - y = 2 2y - 6x = -4 (c) 2x - 3y = 5 3y - 2x = 2
Find all solutions (if any) to each of the following systems of linear equations. (a) x + y + 2z = 8 3x - y + z = 0 -x + 3y + 4z = -4 (b) -2x + 3y + 3z = -9 3x - 4y + z = 5 -5x + 7y + 2z = -14 (c) x
Express the last equation of each system as a sum of multiples of the first two equations. (a) x1 + x2 + x3 = 1 2x1 - x2 + 3x3 = 3 x1 - 2x2 + 2x3 = 2 (b) x1 + 2x2 - 3x3 = -3 x1 + 3x2 - 5x3 = 5 x1 -
Find all solutions to the following systems. (a) 3x1 + 8x2 - 3x3 - 14x4 = 2 2x1 + 3x2 - x3 - 2x4 = 1 x1 - 2x2 + x3 + 10x4 = 0 x1 + 5x2 - 2x3 - 12x4 = 1 (b) x1 - x2 + x3 - x4 = 0 -x1 + x2 + x3 + x4 =
In each of the follow ing, find conditions on a and b such that the system has no solution, one solution, and infinitely many solutions. (a) x - 2y = 1 ax + by = 5 (b) x + by = -1 ax + 2y = 5 (c) x
In each of the following, find (if possible) conditions on a, b, and c such that the system has no solution, one solution, or infinitely many solutions. (a) 3x + y - z = a x - y + 2z = b 5x + 3y -
Consider the following statements about a system of linear equations with augmented matrix A. In each case either prove the statement or give an example for which it is false. (a) If the system is
Consider a homogeneous system of linear equations in n variables, and suppose that the augmented matrix has rank r. Show that the system has nontrivial solutions if and only if n > r.
In each of the following, find all values of a for which the system has nontrivial solutions, and determine all solutions in each case. (a) x - 2y + z = 0 x + ay - 3z = 0 -x + 6y - 5z = 0 (b) x +
In each case, either write V as a linear combination of X, Y, and Z, or show that it is not such a linear combination.Let(a) (b) (c) (d)
In each case, either express Y as a linear combination of the Aj, or show that it is not such a linear combination. (a) T = [l 2 4 O]T A1 = [-13 0 1]T A2 = [3 1 2 0]T A3 = [ 1 1 1 I]T (b) Y = [-l
For each of the following homogeneous systems, find a set of basic solutions and express the general solution as a linear combination of these basic solutions. (a) x1 + 2x2 - x3 + 2x4 +x5=0 x1 + 2x2
In each case determine how many solutions (and how many parameters) are possible for a homogeneous system of four linear equations in six variables with augmented matrix A. Assume that A has nonzero
(a) Show that there is a line through any pair of points in the plane. (b) Generalize and show that there is a plane ax + by + cz + d = 0 through any three points in space.
Find the possible flows in each of the following networks of pipes.(a)(b)
A proposed network of irrigation canals is described in the accompanying diagram.At peak demand, the flows at interchanges A, B, C, and D are as shown.(a) Find the possible flows. (b) If canal BC is
NH3 + CuO → N2 + Cu + H20. Here NH3 is ammonia, CuO is copper oxide, Cu is copper, and N2 is nitrogen.
Find all solutions to the following systems of linear equation. (a) x1 + 4x2 - x3 + x4 = 2 3x1 + 2x2 + x3 + 2x4 = 5 x1 - 6x2 + 3x3 =1 x1+ 14x2 - 5x3 + 2x4 = 3
In each case find (if possible) conditions on a, b, and c such that the system has zero, one, or infinitely many solutions. (a) x + 2y - 4z = 4 3a - y + 13z = 2 4x + y + a2z = a + 3 (b) x + y + 3z =
Show that any two rows of a matrix can be interchanged by elementary row transformations of the other two types.
Find a, b, and c so that the system x + ay + cz = 0 bx + cy - 3z = 1 ax: + 2y + bz = 5 has the solution x = 3,y = -1, z = 2.
Show that the real systemhas a complex solution: x = 2, y = i, z = 3 - i where i2 = -1. Explain. What happens when such a real system has a unique solution?
Find a, b, c and d if(a)(b) (c) (d)
If A denotes an m × n matrix, show that A = - A if and only if A = 0.
A square matrix is called a diagonal matrix if all the entries off the main diagonal are zero. If A and B are diagonal matrices, show that the following matrices are also diagonal. A - B
In each case determine all s and t such that the given matrix is symmetric:(a)(b)
In each case find the matrix A.(a)(b)
In each case either show that the statement is true or give an example showing it is false. (a) If A + B = 0, then B = 0. (b) A and AT have the same main diagonal for every matrix A. (c) If A and B
Compute the following:(a)(b) [3 - 1 2] - 2[9 3 4] + [3 11 - 6] (c) (d)
Compute the following (where possible).Let(a) 5C (b) B + D (c) (A + C)T (d) A - D
Find A in terms of B if: 2A - B = 5(A + 2B)
If X, Y, A, and B are matrices of the same size, solve the following equations to obtain X and Y in terms of A and B. (a) 4X + 3Y = A 5X + 4Y = B
Find all matrices X and Y such that: 2X - 5Y = [1 2]
Simplify the following expression where A, B, and C are matrices. (b) 5[3(A - B + 2C) - 2(3C - B) - A] + 2[3(3A - B + C) + 2(B - 2A) - 2C]
If A is any 2 Ã— 2 matrix, show that:for some numbers p, q, r, and s.
In each case find a system of equations that is equivalent to the given vector equation. (Do not solve the system.)(a)
In each case either show that the statement is true, or give an example showing that it is false. (a) If AX has a zero entry, then A has a row of zeros. (b) Every linear combination of vectors in Rn
Let T: R2 → R2 be a transformation. In each case show that T is induced by a matrix and find the matrix. (a) T is reflection in the line y = x. (b) T is clockwise rotation through π/2.
Let T: R3 → R3 be a transformation. In each case show that T is induced by a matrix and find the matrix. (a) T is reflection in the y - z plane.
If a vector B is a linear combination of the columns of A, show that the system AX = B is consistent (that is, it has at least one solution.)
Let X1 and X2 be solutions to the homogeneous system AX = 0. Show that tX1 is a solution to AX = 0 for any scalar t.
In each case find a vector equation that is equivalent to the given system of equations. (Do not solve the equation.) (a) x1 - x2 - x3 + x4 = 5 - x1 + x3 - 2x4 = - 3 3x1 - 4x2 + 9x3 - 2x4 = 12
In each case compute AX using:(i) Definition 1.(ii) Theorem 4.(a)(b)
In each case, express every solution of the system as a sum of a specific solution plus a solution of the associated homogeneous system. (a) x - y - 4z = - 4 x + 2y + 5z = 2 x + y + 2z = 0 (b) 2x1 +
If X0 and X1 are solutions to the homogeneous system of equations AX = 0, use Theorem 2 to show that sX0 + tX1 is also a solution for any scalars s and t (called a linear combination of X0 and X1).
In each case write the system in the form AX = B, use the gaussian algorithm to solve the system, and express the solution as a particular solution plus a linear combination of basic solutions to the
Compute the following matrix products.(a)(b) (c) (d) (e)
In each case give formulas for all powers A, A2, A3 ... of A using the block decomposition indicated.(a)
Compute the following using block multiplication (all blocks are k Ã— k).(a)(b) (c)
Let A denote an mx.n matrix. If YA = 0 for every 1 × m matrix Y, show that A = 0.
Simplify the following expression where A, B, and C represent matrices. (a) A(B + C - D) + B(C - A + D) - (A + B)C + (A - B)D
If A and B commute with C, show that the same is true of: kA, k any scalar
In each of the following cases, find all possible products A2, AB, AC, and so on.(a)(b)
If A and B are symmetric, show that AB is symmetric if and only if AB = BA.
Let B be an n x n matrix. Suppose AB = 0 for some nonzero m × n matrix A. Show that no n × n matrix C exists such that BC = I.
For the directed graph at the right, find the adjacency matrix A, compute A3, and determine the number of paths of length 3 from v1 to v4 and from v2 to v3.
In each case either show the statement is true, or give an example showing that it is false. (a) If AJ = A, then J = I. (b) I is symmetric, then I + A is symmetric. (c) If A ≠ 0, then A2 ≠ 0. (d)
(a) If A and B are 2 × 2 matrices whose rows sum to 1, show that the rows of AB also sum to 1. (b) Repeat part (a) for the case where A and B are n × n.

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