Ok, i've never studied it with degree, so mind my ignorance, but have i got the concept of Kernal operation correctly..?

From what i understand, mathematicians can plug the variable K into an equation, where k represents the function of two mathematical systems???? Is this right... and if it is... could the system lets suppose be like this...

x^{2}+y^{2} would be the value of K, so the two mathematical functions are working together

(x+\sqrt{2})(x-\sqrt{2})

Or do i have this wrong..?

Any comments are appreciated.

Reiku, you seem to be burdened by a misapprehension. Depending on which space your in, definitions differ slightly, but if I give a couple of examples you might see a pattern.

Suppose f:S \to T is a function on sets, then x,\;y \in \ker (f) \in S if f(x)=f(y) \in T. This induces an equivalence relation on S by "x is f-related to y".

Where there is a closed binary operation on a set, i.e we are working with monoids, groups, vector spaces, rings,....., the definition is a bit tighter. Suppose \sigma:G \to H is a group homomorphism. Then h \in \ker(\sigma) \in G if \sigma(h) =e_H, the identity on H. This is nice, as it means that \sigma(e_G) \in \ker(\sigma) so it can be a subgroup of G (in fact it's a normal subgroup).

The general definition for vector spaces is similar, where the identity is taken to be the 0-vector.

For an inner product space we have: let g:V \times V \to \mathbb{R} be a bilinear form. Then v \in \ker(g) if g(v,w) = 0 (number zero, note), for fixed v and all w

Thanks you