Does matrix multiplication require an inner product space?
It would seem to me that since multiplying a vector by a matrix is simply a linear map, and multiplying two matrices the composition of a linear maps, these must be defined in any vector space, and should not depend on an inner product.
Yet, any definition of matrix multiplication I can consider immediately induces an inner product. I understand that you can multiply a vector by a matrix without using an inner product, but, once you do so, you immediately define an inner product.
So: Can matrix multiplication exist in vector spaces without inner products? What is the relationship between the two?