$x = (x_1,x_2)$, each is an element of $\Bbb R$
$y = (y_1,y_2)$, each is an element of $\Bbb R$
Show the following defines an inner product in $\Bbb R^2$ or show otherwise:
a) $\langle x,y\rangle = 3x_1x_2-x_1y_2-x_2y_1+3x_2y_2$
b) $\langle x,y\rangle = 3x_1x_2-x_1y_2-x_2y_1-3x_2y_2$
What I have tried so far:
a) Axiom 1 $
= $3x_1(ax_2)-(ax_1)y_2-(ax_2)y_1+3(ax_2)y_2$
= $3ax_1x_2-ax_1y_2-ax_2y_1+3ax_2y_2$
= $a(3x_1x_2-x_1y_2-x_2y_1+3x_2y_2)$
= $a
Axiom 2 $
= $3x_1x_2-x_1(y_2+z)-x_2(y_1+z)+3x_2(y_2+z)$
= $3x_1x_2-x_1y_2-x_1z-x_2y_1-x_2z+3x_2y_2+3x_2z$
= $(3x_1x_2-x_1y_2-x_2y_1+3x_2y_2)+(-x_1z-x_2z+3x_2z)$
= $(3x_1x_2-x_1y_2-x_2y_1+3x_2y_2)+(-x_1z+2x_2z)$ What's going on here?
Axiom 3 $