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Here's how you can apply this interpretation of matrix multiplication:

Think of the columns of the matrix A as basis vectors of a coordinate system represented in global coordinates, and the vector v as the components of a vector in that coordinate system. Then the product A * v transforms the vector v into the global coordinate system.

Carrying this forward, matrix multiplication A * B gives the combined representation of two coordinate transformations.

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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.

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