[soVeryTired]

It's a nice article - you focus on matrices as a kind of operator that takes a vector as input and produces another vector. This is one side of the coin.

The other interpretation is that matrices are functions that take two arguments (a row vector and a column vector) and produce a real number. IMO this interpretation opens the door to deeper mathematics. It links in to the idea that a column vector is a functional on a row vector (and vice versa), giving you the notion of dual space, and ultimately leading on to differential forms. It also makes tensor analysis much more natural in general.

[kaitai]

If you're going to attempt a definition like this you need some more conditions (approaching concepts like linearity, for instance). Otherwise you can have a black box like x^3y^3-u^3v^3 that takes in [u,v] and [x,y]^T and spits out a real number; that's not a matrix-y operation.

[p1esk]

That didn’t make any sense to, and I work with matrices every day. Are you trying to describe a dot product?

[soVeryTired]

I'm describing a somewhat unusual way of thinking about vectors, matrices etc. At least, it's unusual from the perspective of someone with an engineering / CS background.

First think about row and column vectors. A row vector and a column vector can be combined via standard matrix multiplication to produce a real number. From that perspective, a row vector is a function that takes a column vector and returns a real number. Similarly, column vectors take row vectors as arguments and produce real numbers.

It turns out that row (column) vectors are the only linear functions on column (row) vectors. This result is known as the Reisz representation theorem. If I give you a linear function on a row vector, you can find a column vector so that computing my function is equivalent to calculating a matrix multiply with your column vector.

Now on to matrices. Matrices take one row vector and one column vector and produce a real number. I can feed a matrix a single argument - the row vector, say - so that it becomes a function that takes one more argument (the column vector) before it returns a real number. Sort of like currying in functional programming. But as we said, the only linear functions that map column vectors into real numbers are row vectors. So by feeding our matrix one row vector, we've produced another row vector. This is the "matrices transform vectors" perspective in the OP's article. But I think the "Matrices are linear functions" perspective is more general and more powerful.

This perspective of vectors, matrices, etc... as functions might seem needlessly convoluted. But I think it's the right way to think about these objects. Tricky concepts like the tensor product and vector space duality become relatively trivial once you come to see all these objects as functions.

[p1esk]

I appreciate you trying to explain it, however I believe it would have really helped if you started from the examples where this way of thinking is useful: you mentioned tensor product and vector space duality, however unless someone is already familiar with these concepts (I'm not) then it does seem needlessly convoluted. Are there any practical applications of these concepts that you can describe?

[lmkg]

I haven't been involved in abstract math in close to a decade, but I think it's a description of the (general) inner product. So, a generalization of the dot product. The classic dot product is that operation with the identity matrix. My understanding is that using matrices that way is very common in physics.

[p1esk]

Any example of some practical use that would make it easier to understand?