This paper studies the decoupling principle of a linear vector channel, which is an extension of CDMA and MIMO channels. We show that the scalar-channel characterization obtained via the decoupling principle is valid not only for collections of a large number of elements of input vector, as discussed in previous studies, but also for individual elements of input vector, i.e. the linear vector channel for individual elements of channel input vector is decomposed into a bank of independent scalar Gaussian channels in the large-system limit, where dimensions of channel input and output are both sent to infinity while their ratio fixed.