13 ensly3asinedand(a)tedDo(d)endjnit,7. XI11. A =412./A =-413. Let u =9. 3x₁ + x₂ - 5x3 = 9Xx₂ + 4x3 = 0+ x2Cle4ZI8. ²[-2] + ² [ 3 ] + ² [¯ ] + ²[8]-[8]414. Let u =o trIn Exercises 9 and 10, write the system first as a vector equationand then as a matrix equation.0−215. Let A =Given A and b in Exercises 11 and 12, write the augmented matrixfor the linear system that corresponds to the matrix equationAx = b. Then solve the system and write the solution as a vector.1-30-53-5-1241−40442-15Where is u?2-3-[-]-25,b=−3+ x3123and A =u?1b= =780216. Repeat Exercise 15: A =10. 8x₁ - x₂ = 45x₁ + 4x₂ = 1X₁ - 3x₂ = 2spanned by the columns of A? (See the figure.) Why or whynot?OLu?5and A = 06-8=-40-2m293 -5-2 61 10121 911di2 -1= [ ²3 ] and b = [ 1 2 ]-6b₂-3FIs u in the plane R81130Huloodoo sno ni sio 15 nov na ROof R³ spanned by the columns of A? Why or why not?Plane spanned bythe columns of ADRM IAEUROog bns befolab7-1. Is u in the subsetboistolAx = b does not have a solution for all possible b, anddescribe the set of all b for which Ax = b does have asolution.Show that the equation1 -3 -42 6,b=b₁b2--]=b35 -1 -8Exercises 17-20 refer to the matrices A and B below. Makeappropriate calculations that justify your answers and mention anappropriate theorem.A =1-102an300 3-1 -1 1-401.4 The Matrix Equation Ax = b 412-83 -117. How many rows of A contain a pivot position? Does theequation Ax=b have a solution for each b in R4?21. Let v₁ =18. Do the columns of B span R4? Does the equation Bx = yhave a solution for each y in R4?19. Can each vector in R4 be written as a linear combination ofthe columns of the matrix A above? Do the columns of Aspan R4?20. Can every vector in R4 be written as a linear combination ofbe the columns of the matrix B above? Do the columns of Bspan R³?land z brs bouto (osqu Wolon0010-10aoJoue.TOYB =224 Let V₁ = -2, V₂ =1010, V₂ =3 -2211 -52 -3 7-2-8 2 -1외01Does (V1, V2, V3} span R4? Why or why not?of seconds to cloduiya yms yuiesbl nohup0- [9].Does {V1, V2, V3} span R³? Why or why not?0-38, V3 =Arb, V3 =00=In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.4-1-523. a. The equation Ax = b is referred to as a vector equation.b. A vector b is a linear combination of the columns of amatrix A if and only if the equation Ax = b has at leastone solution.not nolsX16Mc. The equation Ax = b is consistent if the augmented ma-trix [ A b] has a pivot position in every row.Comd.The first entry in the product Ax is a sum of products.If the columns of an m x n matrix A span R", then theequation Ax = b is consistent for each b in Rm.f. If A is an m x n matrix and if the equation Ax = b isinconsistent for some b in Rm, then A cannot have a pivoposition in every row.24./a. Every matrix equation Ax = b corresponds to a vectoequation with the same solution set.b. Any linear combination of vectors can always be writtein the form Ax for a suitable matrix A and vector x.c. The solution set of a linear system whose augmentmatrix is [a₁ a2a2 a3a3 b] is the same as the solutiset of Ax=b, if A = [a₁ a2 a3 ].d. If the equation Ax = b is inconsistent, then b is not inset spanned by the columns of A.e. If the augmented matrix [A b] has a pivot positionevery row, then the equation Ax = b is inconsistent.

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13 ensly3asinedand(a)tedDo(d)endjnit,7. XI11. A =412./A =-413. Let u =9. 3x₁ + x₂ - 5x3 = 9Xx₂ + 4x3 = 0+ x2Cle4ZI8. ²[-2] + ² [ 3 ] + ² [¯ ] + ²[8]-[8]414. Let u =o trIn Exercises 9 and 10, write the system first as a vector equationand then as a matrix equation.0−215. Let A =Given A and b in Exercises 11 and 12, write the augmented matrixfor the linear system that corresponds to the matrix equationAx = b. Then solve the system and write the solution as a vector.1-30-53-5-1241−40442-15Where is u?2-3-[-]-25,b=−3+ x3123and A =u?1b= =780216. Repeat Exercise 15: A =10. 8x₁ - x₂ = 45x₁ + 4x₂ = 1X₁ - 3x₂ = 2spanned by the columns of A? (See the figure.) Why or whynot?OLu?5and A = 06-8=-40-2m293 -5-2 61 10121 911di2 -1= [ ²3 ] and b = [ 1 2 ]-6b₂-3FIs u in the plane R81130Huloodoo sno ni sio 15 nov na ROof R³ spanned by the columns of A? Why or why not?Plane spanned bythe columns of ADRM IAEUROog bns befolab7-1. Is u in the subsetboistolAx = b does not have a solution for all possible b, anddescribe the set of all b for which Ax = b does have asolution.Show that the equation1 -3 -42 6,b=b₁b2--]=b35 -1 -8Exercises 17-20 refer to the matrices A and B below. Makeappropriate calculations that justify your answers and mention anappropriate theorem.A =1-102an300 3-1 -1 1-401.4 The Matrix Equation Ax = b 412-83 -117. How many rows of A contain a pivot position? Does theequation Ax=b have a solution for each b in R4?21. Let v₁ =18. Do the columns of B span R4? Does the equation Bx = yhave a solution for each y in R4?19. Can each vector in R4 be written as a linear combination ofthe columns of the matrix A above? Do the columns of Aspan R4?20. Can every vector in R4 be written as a linear combination ofbe the columns of the matrix B above? Do the columns of Bspan R³?land z brs bouto (osqu Wolon0010-10aoJoue.TOYB =224 Let V₁ = -2, V₂ =1010, V₂ =3 -2211 -52 -3 7-2-8 2 -1외01Does (V1, V2, V3} span R4? Why or why not?of seconds to cloduiya yms yuiesbl nohup0- [9].Does {V1, V2, V3} span R³? Why or why not?0-38, V3 =Arb, V3 =00=In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.4-1-523. a. The equation Ax = b is referred to as a vector equation.b. A vector b is a linear combination of the columns of amatrix A if and only if the equation Ax = b has at leastone solution.not nolsX16Mc. The equation Ax = b is consistent if the augmented ma-trix [ A b] has a pivot position in every row.Comd.The first entry in the product Ax is a sum of products.If the columns of an m x n matrix A span R&#34;, then theequation Ax = b is consistent for each b in Rm.f. If A is an m x n matrix and if the equation Ax = b isinconsistent for some b in Rm, then A cannot have a pivoposition in every row.24./a. Every matrix equation Ax = b corresponds to a vectoequation with the same solution set.b. Any linear combination of vectors can always be writtein the form Ax for a suitable matrix A and vector x.c. The solution set of a linear system whose augmentmatrix is [a₁ a2a2 a3a3 b] is the same as the solutiset of Ax=b, if A = [a₁ a2 a3 ].d. If the equation Ax = b is inconsistent, then b is not inset spanned by the columns of A.e. If the augmented matrix [A b] has a pivot positionevery row, then the equation Ax = b is inconsistent.

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13. Let u = A = [4] and 4-[-28] 1 1 6 Is u in the plane R³ spanned by the columns of A? (See the figure.) Why or why not? 01 u? EL 11 -u? * Where is u? ninima od over orrespondok o form JA Plane spanned by the columns of A IS UTAM bns batolab