decorrelating detector

In this paper decorrelating detector we have the decorrelating detector. With $$K$$ users, the correlation matrix of the signature waveform is $$R=S^TS$$ with $$S=[s_1,...,s_K]$$. Moreover $$size(R)=K \times K$$

This is my question: In the article (p $$679$$), we have: $$d_1=\sum_{i=1}^K [R^{-1}]_{1,i}s_i$$. But, how can we make this product, the elements do not have the same dimensions? In fact, $$size(R^{-1})=K \times K$$ and $$size(s_i)=N \times 1$$ with $$N \neq K$$

It all works out because $$[\mathbf{R}^{-1}]_{1, i}$$ is just a number. It is the number in row $$1$$ column $$i$$ of the matrix $$\mathbf{R}^{-1}$$. So the product $$[\mathbf{R}^{-1}]_{1, i}\mathbf{s}_i$$ returns a vector of length $$N \times 1$$. See the line after Equation 10.