Determine whether each of statements a through f below are true or false. Justify each answer.a. Every matrix equation Ax = b corresponds to a vector equation with the same solution set. Choose the correct answer below.O A. False. The matrix equation Ax = b only corresponds to an inconsistent system of vector equations.O B. True. The matrix equation Ax = b is simply another notation for the vector equation x, a, +x,a, +•.. + X,an =b, where a,, ., a, are the rows of A.O C. False. The matrix equation Ax = b does not correspond to a vector equation with the same solution set.O D. True. The matrix equation Ax = b is simply another notation for the vector equation x, a, +x,a, + •.. + X,an =b, where a,, ., an are the columns of A.b. If the equation Ax = b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below.O A. True. The equation Ax =b has a nonempty solution setand only if b is a linear combination of the columns of A.O B. True. The equation Ax =b has a solution set if and only if A has a pivot position in every row.O C. False. Ax = b is only consistent if the values of b are nonzero.O D. False. b is only included in the set spanned by the columns of A if Ax = b is inconsistent.c. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. Choose the correct answer below.O A. True. The matrix A is the matrix of coefficients of the system of vectors.O B. False. A and x cannot be written as a linear combination because the matrices do not have the same dimensions.O c. True. Ax can be written as a linear combination of vectors because any two vectors can be combined by addition.O D. False. A and x can only be written as a linear combination of vectors if and only if in Ax = b, b is nonzero.d. If the coefficient matrix A has a pivot position in every row, then the equation Ax = binconsistent. Choose the correct answer below.O A. True. If A has a pivot position in every row, then the augmented matrix must have a row of all zeros, indicating an inconsistent system of equations.O B. False. If a coefficient matrix A has a pivot position in every row, then the equation Ax = b may or may not be consistent. O C. True. A pivot position in every row of a matrix indicates an inconsistent system of equations because the augmented column will always be zeros.O D. False. If A has a pivot position in every row, the echelon form of the augmented matrix could not have a row such as [0 00 1], and Ax = b must be consistent.e. The solution set of a linear system whose augmented matrix isa, a, aza, az az bis the same as the solution set of Ax =b, if A=Choose the correct answer below.O A. False, If A is an mxn matrix with columnsa, a2 ... anthen b cannot be found in R", and the system is inconsistent.O B.False. The solution set of a linear system whose augmented matrix isa, az az bis the same as the solution set of Ax = b if and only if x has the same number of rows as A.Ос.True. If A is an mxn matrix with columns, and b is a vector in R", the matrix equation Ax = b has the same solution set as the system of linear equations whose augmented matrix isa, a, •.. an ba, a2 ..•anO D.True. The linear system whose augmented matrix isa, a, az bwill have the same solution set as Ax = b if and only if b is nonzero.f. If A is an mxn matrix whose columns do not span R", then the equation Ax = b is consistent for every b in R". Choose the correct answer below.O A. False. If the columns of A do not span R", then A does not have a pivot position in every row, and row reducing Ab could result in a row of the form o o... 0 c I, where c is a nonzero real number.O B. True. If Ax = b is consistent, then the rows of A must span R"O C. True. If the columns of A do not span R", b may or may not span R".O D. False. If the columns of A do not span Rm, Ax = b cannot be consistent.

Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

a. Every matrix equation Ax = b corresponds to a vector equation with the same solution set. Choose the correct answer below. O A. False. The matrix equation Ax =b only corresponds to an inconsistent system of vector equations. B. True. The matrix equation Ax = b is simply another notation for the vector equation x, a, + X2a2 + •.. + Xnan = b, where a,, ..., an are the rows of A. O C. False. The matrix equation Ax = b does not correspond to a vector equation with the same solution set. O D. True. The matrix equation Ax = b is simply another notation for the vector equation x, a, + x2a, + •.. +Xnan = b, where a,, ., an are the columns of A. b. If the equation Ax = b is consistent, then b is in the set spanned by the columns of A. Choose the correct answer below. O A. True. The equation Ax = b has a nonempty solution set if and only if b is a linear combination of the columns of A. O B. True. The equation Ax = b has a solution set if and only if A has a pivot position in every row. OC. False. Ax = b is only consistent if the values of b are nonzero. O D. False. bis only included in the set spanned by the columns of A if Ax = b is inconsistent. c. Any linear combination of vectors can always be written in the form Ax for a suitable matrix A and vector x. Choose the correct answer below. O A. True. The matrix A is the matrix of coefficients of the system of vectors. B. False. A and x cannot be written as a linear combination because the matrices do not have the same dimensions. OC. True. Ax can be written as a linear combination of vectors because any two vectors can be combined by addition. O D. False. A and x can only be written as a linear combination of vectors if and only if in Ax = b, b is nonzero.

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ