11.32Consider the matrix A =-3-121.-1-241.V.Find the basis and dimension of N(A) (N(A) is the solution space of thehomogeneous system Ax = 0 )vi.If the system Ax = b is consistent where b =find the complete solutionin the form x = x,+xpwhere x, denotes a particular solution and xp denotes a%3Dsolution of the associated nonhomogeneous system Ax = 0.

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Answered: 1 1 3 1 2 Consider the matrix A = -3 -1…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find a basis for and the dimension of the solution space of the homogeneous system of linear equations−x + y + z = 02x − y = 04x − 5y − 6z = 0(a) a basis for the solution space
A tank contains 200 gal of liquid. Initially, the tank contains pure water. At time t = 0, brine containing 3 lb/gal of salt begins to pour into the tank at a rate of 2 gal/min, and the well-stirred mixture is allowed to drain away at the same rate. How many minutes must elapse before there are 100 lb of salt in the tank?
2. Determine a basis for and the dimension of the solution space of the homogeneous system X3 + X4 + X5 = 0 -x1 – x2 + 2x3 – 3x4 + x5 = 0 X1 + x2 – 2x3 - X5 = 0 2х, + 2x2 — Хз + X5
16. 3 [1 1 A=0 1 -2 11 2. 21 (Part1) Let Use elementary row operations to find the inverse A-1 of A. ii) Use your result in (i) above to solve the system (x+y+3z = 1 y-2z 2 x+2y+2z% = 1 i) Solve the system (AT)-1X = 0 (Part2) (1) Express, if possible, the vector t (1,2,-6) as a linear combination of the vectors u= (0, 1, 2)", v = (1,0, 2)", w = (1,2,0)". (11) Let E = (u,, u2, u) and F (b,,b), where u, = (1,0. – 1)",u, = (1, 2, 1)", u, = (-1.1.1)" and b; (1,-1)', b2 = (2,-1)'. Find the matrix representing L from R' into R' with respect %3D %3D to the ordered basis E and F where L(x) = (2x2,-x,).
fast pls
2. (a) Use the linearity of matrix-vector multiplication to prove the following: if the vectors x1, X2 are solutions to Ах = 0 (3.1) then the linear combination x = cịx1 + c2x2 is also a solution, for any constants c1, c2. (You may use the results you showed in the previous problem) (b) Show that the results is not true for nonhomogeneous systems: if the vectors x1, X2 are solutions to Ax = b (3.2) with b + 0, then the linear combination x = c1x1 + c2x2 is not a solution. (c) Show that if, however, x1 and x2 are solutions to the homogenous problem (3.1) and is a particular solution to the nonhomogeneous problem (3.2), then Хр х— С1X1 + C2X2 + Хр is a solution for the nonhomogeneous problem (3.2), for any c1, C2.
Gaussian elimination with partial pivoting is applied to the matrix [2 1 0 3; -5 1 2 0; -2 0 1 2; -4 -4 -5 1] a) m41= -2 b) m41= -2/5 c) m41= 4/5 d) m41= 5/2
** || Find the least-squares solution x 1 1 1 1 of the system 1 1 -1 -1 X -1_1 1 -1] || -11 -9 -5 5
2. A is a 4 x 5 matrix with four pivot positions. a) Does the equation Ax = 0 have a nontrivial solution? b) Does the equation Ax = b have at least one solution for every possible b?
(a) Use Gauss-Jordan elimination to find the inverse of the given matrix A 1 1 3 3 -1 1 A = -1 -3 -4 2 -5 2 (b) Given b = (1.1.2.2) solve the system by suing x = A-'b.
5
2. The augmented matrix for the system of equations 211 + 2r2 + I3 = 5, 2i + 3r2 + a3 = 2, 3r1 + 4r2 + 5z3 = 1, is given by O (2 2 1 1 4 5 2x 2x2 3r2 3z, 4r2 5z3 1 2 2 1 3. 1 2 4 5 1 '5 2 2 1 2. 3 4 5 1 1 1 213 213

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