Students try to compare Matrix multiplication to normal multiplication first time they hear or see matrix multiplication and they get lost without developing any intuition for what does it actually mean.
Now everybody knows that normal multiplication b*a just means adding a for b times, don’t get me started with the case of non-integers here, because we will deal with that in other article.
Normal multiplication is kind of dealing with vector(a) in one dimension i.e. straight line. You can go anywhere (or span) the whole line just by multiplying it with varied values of b.
Suppose vector(a) is two dimensional of sort(xi+yj). Since the vector is two dimensional by it’s very nature, it has the potential to span whole two dimensional space if we multiply it by right kind of b. If b is scalar, then you are stuck in one dimension i.e. in the direction of xi+yj. If b is a two dimensional vector, then multiplying b with col(xi yj) gives you another column vector. ooh.. slow down buddy.. what has happened just now..Think of two columns of two dimensional vector as two vectors, if x and y or 1,2 then you take 1 times first column vector, then 2 times second column vector and add both of them, result is another column vector. This is called linear combination. This is not really different from the concept of vector where you take x times in i direction and y times in j direction to end up at a point in 2-d except for the fact that you are taking x times in 1st column vector direction and y times in 2nd column vector direction. Matrix multiplication is Linear Transformation or transformations based on the size of matrices being multiplied.
Think of column vectors of the multiplication matrix , often left side one in A*B , as the places where original i,j,k.. unit vectors of your n dimensional space end up after matrix multiplication. Now think of your multiplication result of A*B as the places where column vectors of B end up after the Linear transformation or Matrix multiplication.
Matrix Multiplication is non-abelian i.e. AB!=BA
Hope this is helpful..Correct me if I am wrong.