Another way to look at it: A matrix is a linear transformation, and multiplying a vector by a matrix is how you apply the transformation. But linear transforma

Another way to look at it: A matrix is a linear transformation, and multiplying a vector by a matrix is how you apply the transformation. But linear transformations are really just changes of basis. How do you change your basis? You find the dot product of a vector with each new basis vector. And that's exactly what matrix multiplication is. When you multiply your column vector by a row in the matrix, you're finding the dot product, doing the projection in your change of basis.