(a) Prove that if A € M22(F), then det(A) + 0 if and only if the columns of A are linearly independent. ) Define \mathrm{GL}(\mathbf{2}, \mathbf{F})=\left\{A \in M_{2,2}(\mathbf{F}) \mid \operatorname{det}(A)

\neq 0\right\} . and recall that GL(2, F) is a group with the operation of matrix multiplication. Use (a)to calculate |GL(2, IF)|. Define D = {A € GL(2, F) | A is a diagonal matrix} ; T = {A € GL(2, F) | A is an upper triangular matrix} . Prove that T is a subgroup of GL(2, F); prove that D is a subgroup of T; calculate |D|and calculate |T|. 1) Using the groups D and T from (c), consider the following function Prove that y is a homomorphism and calculate ker(9) and state its order. Let F = {0, 1, ...,p-1} be the field of order p (where p is a prime, and we perform arithmetic modulo p).