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".