Matrix multiplication is clearly defined but is there a tangible or physical interpretation for it? Or is it just abstract formalism? I am thinking about each column of the matrix as vectors so matrix multiplication with two 2by2 matrices is about multiplying 4 vectors in a certain way. The result is a 2by2 matrix with all four vectors tangled up with each other. But does the two resultant vectors in this matrix have a meaning with respect to the original vectors if we gave the original vectors some meaning in the first place? I do see how an nbyn matrix multiplied with a nby1 matrix is. It is like dot product on a larger scale with one vector dotted with a set of vectors. So the question is about multidimentional matrix multiplication.