I am learning linear algebra for a machine learning class and have a question about matrix multiplication. The product of two matrices is undefined whenever the rows of the first matrix (reading right to left) do not match the column of the second matrix.
However, say that I would need (for some reason) to multiply a 3x3 matrix by a 2x2 one. Couldn't I just complete the operation by adding the "missing row" with coordinates [0,0]? I am thinking about it because if matrices represent linear transformations of space, then a 2x2 matrix represent a two dimensional transformation. However, isn't a two dimensional transformation simply a transformation where every other dimension is equal to 0?
To illustrate that, let's say I want to apply a linear transformation [-1,0;0,1] to vector [3,3]. The resulting vector would be a two dimensional vector [-3,3]. Now let's say that after this transformation I want to apply another transformation to the same vector, but this time in three dimensions. To keep the example as simple as possible let's use the identity transformation for this: [1,0,0;0,1,0;0,0,1]. Doing this would require to multiply the 3x3 matrix [1,0,0;0,1,0;0,0,1] by the 2x3 one [-1,0;0,1], which is technically not possible. However, if I apply the method above (i.e. adding a third row with all 0 to the 2x3 matrix) I would still be able to compute the transformation and get the result [-1,0,0;0,1,0]. I can then multiply this to my original vector and get [-3,3,0]. The only difference I can see between [-3,3,0] and [-3,3] is that in the first one I am just "explicitly showing" the third dimension as having coordinate 0, whilst in the second I am keeping this implicit.
What am I missing here?