This is why I'm a big fan of Axler's Linear Algebra Done Right. The book's emphasis is on the concepts behind the calculations rather than the calculations.
I actually disagree with Axler on his avoidance of the determinant, though. I wish instead of avoiding it he'd spent more time developing it conceptually, as it's actually a fascinating construction. But to this day I have yet to find a gentler and better introduction to serious linear algebra than his book.
He recommends it as a second course, but I read it during my first course in the subject and considered it my "secret weapon". I truly believe that book is what allowed me to get a perfect score in the class -- I had a conceptual understanding that was just not possible to glean from the official course textbook.
Axler's book is very good. An alternative to consider is Paul Halmos's much older Finite-Dimensional Vector Spaces [0]. Both books take the same basic approach, and the proofs of the major theorems are substantially the same. (Halmos's book is well known and well liked and was probably Axler's starting point.)
Axler's book covers more ground (most notably, Halmos presents the polar decomposition but not the singular-value decompostion) and uses more modern terminology and notation. But Halmos's book has the merits of being half as long and a third as expensive, as well as having been written specifically to prepare the reader as directly as possible for Halmos's short introduction to Hilbert spaces [1].
I highly recommend one or the other of these books for readers who want to understand linear algebra as mathematicians do.
For those who already know enough linear algebra to be bored with an entire book dedicated to Axler's approach to linear algebra, you can simply read the paper upon which the book is based:
This is why I'm a big fan of Axler's Linear Algebra Done Right. The book's emphasis is on the concepts behind the calculations rather than the calculations.
I could never learn mathematics that way, and goodness knows, school systems in both Europe and America have tried to teach it to me that way. What I understand of mathematics today has mostly been achieved through autodidaction.
Some of us cannot, repeat can not go from the abstract to the empirical; I'm one of those people, who have to go from the empirical to the abstract instead. I need to see the mechanics of it, because I learn by observing the pattern. If I cannot make out the pattern, I cannot comprehend the abstraction.
Take UNIX manual pages for example: the first thing I go to after reading the SYNOPSIS is the EXAMPLES section (and since most GNU/Linux manual pages have no EXAMPLES sections, it's a product which is useless to me, unlike UNIX). The more examples the EXAMPLES section entails, the faster I'm able to grasp the concept. Same with http://matrixmultiplication.xyz/, if I could have learned matrix multiplication that way, visually, instead of having to do it in my head, it would have been far easier. The other way around, it was a slow, painful torture, and to this present day, I cannot multiply the matrices in my head, or without these visualisations to remind me of the rules: they're too complicated for me to keep in my head.
The point I'm trying to make is that not everyone learns the same way, and our brains do not process the information in the same way, even if the final result, understanding of the concept, is identical. A lot of abstract concepts which are difficult to reliably repeat, or whose outcome is either non-deterministic or unclear, we do not even understand the same way, hence opinions often differ by a small or large margin. Don't assume that the learning technique which works for you would work for someone else. That's a major failure of most pedagogical approaches, with the exception of perhaps Montessori and Fröbel. Unlike Montessori and Fröbel, most pedagogical approaches force a uniform way of learning, except that a human brain does not learn in a uniform manner, differing from individual to individual instead. Teaching should be a highly individualized approach.
But surely these sort of meaningless examples with randomized numbers like at the site aren't that useful? I mean, a "real-world" example of matrix multiplication would be having a proper translation or rotation matrix and then visualizing how it maps a point to its image.
That demonstrates multiplication of a matrix with a vector which is being rotated, but multiplication of two matrices corresponds to composition of the corresponding operators.
This is actually pretty instructive. For instance, rotations in three-dimensional space do not commute in general; hence neither does matrix multiplication.
Technically matrix-vector multiplication is just a special case of matrix-matrix-multiplication, but I take your point. Indeed it would be illuminating to see how exactly matrix multiplication composes two transformations.
I can see why he discourages it. It prevents understanding why a matrix is singular. I often find students who use it as the main justification for singularity without further understanding singularity -- namely that it occurs when a linear map sends things to a lower dimensional space than it accepts.
I actually disagree with Axler on his avoidance of the determinant, though. I wish instead of avoiding it he'd spent more time developing it conceptually, as it's actually a fascinating construction. But to this day I have yet to find a gentler and better introduction to serious linear algebra than his book.
He recommends it as a second course, but I read it during my first course in the subject and considered it my "secret weapon". I truly believe that book is what allowed me to get a perfect score in the class -- I had a conceptual understanding that was just not possible to glean from the official course textbook.