Final answer:
The correct answer is that a. G ∘ F is a linear transformation with standard matrix AB, as the composition of two linear transformations results in a matrix obtained by multiplying the matrices of the individual transformations in the order they are applied.
Step-by-step explanation:
The student is asking about the standard matrix representation of compositions of linear transformations. Given linear transformations F:R_4 → R_4 and G:R_4 → R_3, with standard matrices A and B respectively, the student needs to identify the correct statement about the composition of these transformations and their standard matrix.
Option (a) states that G ∘ F is a linear transformation with standard matrix AB. This is the correct answer because when you compose two linear transformations, the standard matrix of this composition is the product of the matrices of the individual transformations, in the order they are applied.
Therefore, the matrix for G ∘ F is obtained by multiplying B (matrix for G) with A (matrix for F).
Options (b), (c), and (d) suggest other combinations and orders of matrix multiplication which are incorrect based on the definitions of the transformations given and the order in which they are composed.