I am currently dealing with calculations done on vectors and matrices. For some parts I have gained an intuitive understanding, for others I didn't.
E.g., when we are adding two vectors, you can imagine that this means adding two arrows. The result is a single arrow that reflects the combined forces of the two individual source vectors. The resulting vector will probably have a new direction, which is influenced by the two original ones.
When we multiply a vector by a scalar, you can imagine that this means putting the very same arrow multiple times behind itself, to make it longer. I.e., the new vector has more force, but the direction stay the same.
Now… if I want to multiply a matrix by a vector, what is the analogy for that? What does this mean in terms of geometry?