25 c. The solution set of Axb is obtained by translating thesolution set of Ax = 0,25. Prove the second part of Theorem 6: Let w be any solution ofAxb, and define vw-p. Show that v, is a solutionof Ax 0. This shows that every solution of Axb has theform w p + V. with p a particular solution of Ax = bandVh a solution of Ax = 0.26. Suppose Ax=b has a solution. Explain why the solution isunique precisely when Ax 0 has only the trivial solution.27. Suppose A is the 3 x 3 zero matrix (with all zero entries).Describe the solution set of the equation Ax = 0.28. If b 0, can the solution set of Axb be a plane throughthe origin? Explain.In Exercises 29-32,, (a) does the equation Ax=0 have a nontriv-ial solution and (b) does the equation Ax = b have at least onesolution for every possible b?29. A is a 3 x 3 matrix with three pivot positions.30. A is a 3 x 3 matrix with two pivot positions.31. A is a 3 x 2 matrix with two pivot positions.32. A is a 2 x 4 matrix with two pivot positions.-2 -67 21-3 -933. Given A =find one nontrivial solution ofAx = 0 by inspection. [Hint: Think of the equation Ax = 0written as a vector equation.]RIN34, Given AS124 -5-1 81.5 Solution Sets of Linear Systems to6Ax = 0 by inspection.35. Construct a 3x3 nonzero mara A such that the vesker036. Construct a 3 x 3 monzere matrix A us that the vezzOT12X3is a solution of Ax = 0,137. Construct a 2 x 2 matrix A such that the option set of heequation Az = 0) is the fine in R² through (4.1) and theorigin. Then, find a vector b in R² such that the solution setof Ax=b is not a line in 22 parallel to the solution set ofAx = 0. Why does this not contradict Theorems 6?SOLUTIONS TO PRACTICE PROBLEMS1. Row reduce the augmented matrix:09]-[8And one now ation ofis a solution of Ax = 0.38. Suppose A is a 3 x 3 matrix and y is a vector in ³ such thatthe equation Ax = y does not have a solution. Does thereexist a vector z in R' such that the equation Ax = za aunique solution? Discuss.=39. Let A be an m x matrix and let u be a vector in asas-fies the equation Ax = 0. Show that for any scalare, the wec-tor cu also satisfies Ax = 0. [That is, show that A(cu) = 0.140. Let A be an m x n matrix, and let u and y be vectors in Rºwith the property that Au = 0 and Av = 0. Explain whyA(u + v) most be the zero vector. Then explain whyA(cu + dv) = 0 for each pair of scalars c and d.4-5 0180-9Thus x₁ = 4-3x3, x₂ = −1+2x3, with x3 free. The general solution in parametricx2vector form is4 - 3x3-1 + 2x3X30 3 49]-[14]1 -2 -1+ 3x3 = 4X2 - 2x3 = -1=42[]+[]x304PThe intersection of the two planes is the line through p in the direction of v.

Answered: Image /qna-images/answer/7d217c93-e96a-4cb6-8b69-d3e30c86950d.jpg

e. The solution solution set of Ax = 0. 25. Prove the second part of Theorem 6: Let w be any solution of Axb, and define v, w -- p. Show that v, is a solution of Ax 0. This shows that every solution of Axb has the form wa p + V. with p a particular solution of Ax= b and vi, a solution of Ax = 0. 26. Suppose Axb has a solution. Explain why the solution is han anly the trivial solution.