Definition talk:Quadratic Form (Linear Algebra)

There has to be a way to integrate the definition as a homogeneous quadratic polynomial. They're intimately related, but not exactly the same. --barto (talk) 14:57, 2 May 2017 (EDT)

The subtlety is: often quadratic forms are only defined on $\mathbb K^n$, but eventually one talks about restrictions to arbitrary subspaces, so the cleanest way is without doubt to allow an arbitrary (finite dimensional, if you want) space in the definition. --barto (talk) 15:22, 2 May 2017 (EDT)

The danger is in making the definition so abstract and general that, despite all the simple examples in the world, it becomes impossible to disentangle intuitively the gist of the concept. This is the problem we had with polynomials, and it may be the problem we are going to have here (which I possibly started by going into it at a higher level than advisable).

So however we go about doing it (don't ask me, I'm no expert, all I know here is what I picked up by floundering my way through to chapter 30 of Warner some 10 years or so ago) we need to establish a subdefinition where the underlying field (or cartesian product of fields) is based on the reals or complex numbers, so that less high-flying students have a chance of grasping it. --prime mover (talk) 15:33, 2 May 2017 (EDT)

Sure. What we can do is define quadratic forms on $\mathbb K^n$ via their matrix representation, and then show that these definitions are equivalent (they are, because we're working with free modules). --barto (talk) 15:38, 2 May 2017 (EDT)