Good intuitive introduction to matrix mathematics

I am a post doc of biology, and am supposed to be teaching an undergrad course in biological mathematics. The course has a subsection on matrix operations. My problem is that, though I am familiar in principle with the basic arithmetic operations on matrices, I do not have a good intuitive understanding of them. Could anyone in this forum direct me to a good intuitive introductory text on matrix operations, either on the net or in book form?

Matrix maths has so many real-world applications, it must be possible to get an intuitive grasp of, e.g. what it means to premultiply a matrix by its transpose. It seems to me that most mathematical texts are centered around proofs and axioms, but my poor biological brain simply cannot cope with that. I would love, e.g., a visual geometric representation.

Thanks a lot!

P.S. My sincere apologies if this ended up in the wrong subforum.