Matrix is also orthogonal if $A^{T}A= I_{n}$ Also, regarding change of basis problems, $x= S*[x]_{B}$ where S holds the basis vectors [v1 v2] and $[x]_{B}$ describes the location of x in terms of the basis vectors ( $x = c_{1}v_{1} + c_{2}v_{2} , [x]_{B} = [c_{1} c_{2}]$ as vertical column vector ). With this as well, $[T(x)]_{B} * S = T(x)$