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Mark each statement True or False and justify each answer for parts a through e. a. If A is a 4x 3 matrix, then the transformation x → Ax maps R³ onto Rª. Choose the correct answer below. O A. True. The the columns of A are linearly independent. OB. True. The columns of A span Rª OC. False. The columns of A do not span R4. O D. False. The columns of A are not linearly independent. b. Every linear transformation from R to R is a matrix transformation. Choose the correct answer below. OA. True. Every matrix transformation spans R". O B. False. Not every vector x in R" can be assigned to a vector T(x) in Rm. (...) O C. True. There exists a unique matrix A such that T(x) = Ax for all x in R". O D. False. Not every image T(x) is of the form Ax. c. The columns of the standard matrix for a linear transformation from R to R are the images of the columns of the nxn identity matrix under T. Choose the correct answer below. O A. True. The standard matrix is the mxn matrix whose jth column is the vector T (e), where e; is the jth column of the identity matrix in R OB. False. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e; is the jth column whose entries are all 0. O C. False. The standard matrix only has the trivial solution. O D. True. The standard matrix is the identity matrix in R". d. A mapping T: R^→R" is one-to-one if each vector in R" maps onto a unique vector in R. Choose the correct answer below. O A. False. A mapping T is said to be one-to-one if each b in RM is the image of at least one x in R". O B. True. A mapping T is said to be one-to-one if each x in R" has at least one image for b in Rm. OC. True. A mapping T is said to be one-to-one if each b in RM is the image of exactly one x in R". O D. False. A mapping T is said to be one-to-one if each b in RM is the image of at most one x in R".

Mark each statement True or False and justify each answer for parts a through e. a. If A is a 4x 3 matrix, then the transformation x → Ax maps R³ onto Rª. Choose the correct answer below. O A. True. The the columns of A are linearly independent. OB. True. The columns of A span Rª OC. False. The columns of A do not span R4. O D. False. The columns of A are not linearly independent. b. Every linear transformation from R to R is a matrix transformation. Choose the correct answer below. OA. True. Every matrix transformation spans R". O B. False. Not every vector x in R" can be assigned to a vector T(x) in Rm. (...) O C. True. There exists a unique matrix A such that T(x) = Ax for all x in R". O D. False. Not every image T(x) is of the form Ax. c. The columns of the standard matrix for a linear transformation from R to R are the images of the columns of the nxn identity matrix under T. Choose the correct answer below. O A. True. The standard matrix is the mxn matrix whose jth column is the vector T (e), where e; is the jth column of the identity matrix in R OB. False. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e; is the jth column whose entries are all 0. O C. False. The standard matrix only has the trivial solution. O D. True. The standard matrix is the identity matrix in R". d. A mapping T: R^→R" is one-to-one if each vector in R" maps onto a unique vector in R. Choose the correct answer below. O A. False. A mapping T is said to be one-to-one if each b in RM is the image of at least one x in R". O B. True. A mapping T is said to be one-to-one if each x in R" has at least one image for b in Rm. OC. True. A mapping T is said to be one-to-one if each b in RM is the image of exactly one x in R". O D. False. A mapping T is said to be one-to-one if each b in RM is the image of at most one x in R".

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.8: Determinants
Problem 30E
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Mark each statement True or False and justify each answer for parts a through e. a. If A is a 4x 3 matrix, then the transformation x → Ax maps R³ onto Rª. Choose the correct answer below. O A. True. The the columns of A are linearly independent. OB. True. The columns of A span Rª OC. False. The columns of A do not span R4. O D. False. The columns of A are not linearly independent. b. Every linear transformation from R to R is a matrix transformation. Choose the correct answer below. (...) O A. True. Every matrix transformation spans R". O B. False. Not every vector x in R" can be assigned to a vector T(x) in R OC. True. There exists a unique matrix A such that T(x) = Ax for all x in R". O D. False. Not every image T(x) is of the form Ax. c. The columns of the standard matrix for a linear transformation from R to R are the images of the columns of the nxn identity matrix under T. Choose the correct answer below. OA. True. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e; is the jth column of the identity matrix in R. O B. False. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e, is the jth column whose entries are all 0. False. The standard matrix only has the trivial solution. O C. O D. True. The standard matrix is the identity matrix in R". d. A mapping T: R^→R" is one-to-one if each vector in R" maps onto a unique vector in R. Choose the correct answer below. O A. False. A mapping T is said to be one-to-one if each b in RM is the image of at least one x in R". OB. True. A mapping T is said to be one-to-one if each x in R" has at least one image for b in Rm OC. True. A mapping T is said to be one-to-one if each b in RM is the image of exactly one x in R". O D. False. A mapping T is said to be one-to-one if each b in RM is the image of at most one x in R".
Transcribed Image Text:Mark each statement True or False and justify each answer for parts a through e. a. If A is a 4x 3 matrix, then the transformation x → Ax maps R³ onto Rª. Choose the correct answer below. O A. True. The the columns of A are linearly independent. OB. True. The columns of A span Rª OC. False. The columns of A do not span R4. O D. False. The columns of A are not linearly independent. b. Every linear transformation from R to R is a matrix transformation. Choose the correct answer below. (...) O A. True. Every matrix transformation spans R". O B. False. Not every vector x in R" can be assigned to a vector T(x) in R OC. True. There exists a unique matrix A such that T(x) = Ax for all x in R". O D. False. Not every image T(x) is of the form Ax. c. The columns of the standard matrix for a linear transformation from R to R are the images of the columns of the nxn identity matrix under T. Choose the correct answer below. OA. True. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e; is the jth column of the identity matrix in R. O B. False. The standard matrix is the mxn matrix whose jth column is the vector T (e;), where e, is the jth column whose entries are all 0. False. The standard matrix only has the trivial solution. O C. O D. True. The standard matrix is the identity matrix in R". d. A mapping T: R^→R" is one-to-one if each vector in R" maps onto a unique vector in R. Choose the correct answer below. O A. False. A mapping T is said to be one-to-one if each b in RM is the image of at least one x in R". OB. True. A mapping T is said to be one-to-one if each x in R" has at least one image for b in Rm OC. True. A mapping T is said to be one-to-one if each b in RM is the image of exactly one x in R". O D. False. A mapping T is said to be one-to-one if each b in RM is the image of at most one x in R".
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