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.
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.
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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Question

Transcribed Image Text: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 set
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.
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 = b
inconsistent. 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 is
a, a, az
a, az az b
is 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 columns
a, a2 ... an
then b cannot be found in R", and the system is inconsistent.
O B.
False. The solution set of a linear system whose augmented matrix is
a, az az b
is 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 is
a, a, •.. an b
a, a2 ..•
an
O D.
True. The linear system whose augmented matrix is
a, a, az b
will 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2F74319c2c-0a74-4eb8-bcc7-fc5345ddd9e5%2Fzismw2_processed.png&w=3840&q=75)
Transcribed Image Text: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 is
a, a, az
a, az az b
is 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 columns
a, a2 ... an
then b cannot be found in R", and the system is inconsistent.
O B.
False. The solution set of a linear system whose augmented matrix is
a, az az b
is 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 is
a, a, •.. an b
a, a2 ..•
an
O D.
True. The linear system whose augmented matrix is
a, a, az b
will 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.
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