Viewing C 4 and M2,2(C) as vector spaces over C with the usual vector space operations,
the set
B =
1 0
0 1
,

1 0
0 −1

,

0 1
1 0
,

0 −i
i 0

is a basis for M2,2(C) (in fact, B \ {I2} are called the Pauli matrices :D). Suppose T : M2,2(C) → C
4
is
a linear map such that
T
1 0
0 1 = (2, 1, 0, −1) T
1 0
0 −1
= (0, 1, 2, −1)
T
0 1
1 0 = (0, 3, 0, −3) T
0 −i
i 0
= (0, −1, 0, 1).
Determine the value of
T
z w
u v
for all z, w, u, v ∈ C.