# Linear Algebra - Show that this defines an inner product?

## 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??

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HallsofIvy
Homework Helper
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
Homework Helper
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