Transposition of first matrix in crossprod in R Is there a statistically important reason that the first matrix in crossprod is transposed? That is, why is crossprod(x,y) equivalent to t(x) %*% y, and not just to x %*% y?
 A: As you indicated, %*% already does multiplication; there's not really a need for second function to do the same job. The function crossprod is a shortcut to transpose multiply, both in syntax and computation. For example, the normal equations $\hat{\beta}=(X^TX)^{-1}X^Ty$ in R are solve(crossprod(X), crossprod(X,y)),* (single-argument crossprod computes $X^TX$). It's also faster than solve(t(X)%*%X, t(X)%*%y) because the second expression has to transpose and then multiply; this can be slow. 
The normal equations are just a prominent example of the many, many occasions in statistics where transpose multiply is desired. It's exceptionally convenient to have access to have efficient access to that functionality.
*But you should not directly apply the normal equations for numerical reasons. Use lm, which applies the QR decomposition by default and is more robust to ill-conditioned matrices.
This monograph defines crossproduct in a manner consistent with R's. (Herve Abdi, Lynne J. Williams. "Matrix Algebra." In Neil Salkind (Ed.), Encyclopedia of Research Design. Thousand Oaks, CA: Sage. 2010.)
A: Because it assumes matrices dimensions $m\times n$ so you need to be multiplying $n\times m \cdot m \times n$ to yield a relevant matrix.
