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".
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Answer all the parts otherwise don't solve.
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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