Bilinear property of dot-products I didn't understand the bilinear property of dot products. Is there a property of just being linear as well of some operation? 
 A: Bilinearity is a property of functions that take as arguments two vectors. More specifically, a map $B : V \times V \to \mathbb{R}$ is bilinear if $v \mapsto B(v,w)$ is linear for each $w \in V$, and $w \mapsto B(v,w)$ is linear for each $v \in V$.
It does not make sense (without further clarification) to ask if a function $B:V \times V \to \mathbb{R}$ of two arguments is linear, since linearity is defined for functions that take only one vector as an argument.
A: Not exactly, no. The natural way to think of linearity for, say $f: X \times Y \to \mathbb R$ is the preservation of scalar multiplication in both variables: 
$$ f(\alpha x , \alpha y) = \alpha f (x,y) $$
along with the property of preserving vector addition.
However, if $f$ were bilinear, then:
$$ f(\alpha x , \alpha y) = \alpha^2 f(x,y) $$
That is, bilinear maps are not linear, if you define linearity in the way suggested above. (Though I do struggle to think of alternative definitions that aren't at least a little contrived.)
