Linear Algebra - Show that this defines an inner product?

[proctortom]

Homework Statement

If x = (x1, x2) and y = (y1, y2)...

Show that <x,y> = 3(x1)(y1) - (x1)(y2) - (x2)(y1) + 3(x2)(y2)

Homework Equations

I know that to define it as an inner product space, the following must be correct:

<x,y> = <y,x>
a<x,y> = <ax,y>
<x,y+z> = <x,y> + <x,z>
<x,x> >/= 0
<x,x> = 0 therefore x=0

The Attempt at a Solution

I have a fair idea what the rules above mean, however I have no clue how to apply it to the question. Help??

 

[HallsofIvy]

Replace each <x, y> with the formula you are given, 3x_1y_1- x_1y_2 - x_2y_1+ 3x_2y_2, and see if they are true.

The first one, <x, y>= <y, x> would become 3x_1y_1- x_1y_2 - x_2y_1+ 3x_2y_2= 3y_1x1- y_1x_2 - y_2x_1+ 3y_2x_2. Is that true?

 

[ehild]

In the definition of the above inner product, x1,2, y1,y2 are ordinary real numbers and (x1)(y1)... are products of these numbers.

What do you get if you exchange x and y? As an example, let be (x1,x2)=(2,3) and (y1,y2)=(4,5). What is the inner product <x,y>?
Now let be (x1,x2)=(4,5) and (y1,y2)=(2,3). What is the inner product now?

ehild