Let me consider part b).

The notation just means that maps a typical vector in to the vector , also in .

The matrix of this transformation with respect to the standard basis for (which is (1,0,0,....,0), (0,1,0,....,0), ...., (0,0,...,0,1)) is the matrix, whose first column is the coordinates of T applied to the first basis vector, and similarly for the other columns. To be concrete, T applied to the first basis vector is:

T(1,0,...,0) = (0,0,...,1).

Hence the first column of the matrix is

0

0

.

.

.

0

1

Similarly for the other columns, so the matrix will look something like this:

0 0 . . . 0 1

0 0 . . . 1 0

0 0 . . . 0 0

. . . . . . . .

. . . . . . . .

. . . . . . . .

0 1 . . . 0 0

1 0 . . . 0 0