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Linear Algebra - Show that this defines an inner product?

  1. Oct 14, 2012 #1
    1. The problem statement, all variables and given/known data

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

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

    2. Relevant 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

    3. 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??
     
  2. jcsd
  3. Oct 14, 2012 #2

    HallsofIvy

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

    The first one, <x, y>= <y, x> would become [itex]3x_1y_1- x_1y_2 - x_2y_1+ 3x_2y_2= 3y_1x1- y_1x_2 - y_2x_1+ 3y_2x_2[/itex]. Is that true?
     
  4. Oct 14, 2012 #3

    ehild

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    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
     
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