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Linear Algebra define scalar products

  1. Mar 1, 2015 #1
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    1. The problem statement, all variables and given/known data
    How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation?

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Mar 1, 2015 #2

    haruspex

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    Assuming the 7 is a typo, it's a matter of definition.
     
  4. Mar 1, 2015 #3

    Stephen Tashi

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    Your question isn't clear. Try rephrasing it. Specify the topic you are talking about. Are you asking something about "inner product" or "dot product"?
     
  5. Mar 1, 2015 #4
    The question I was given on my Linear Algebra home assignment is as follows
    Witch of the following formulas define scalar products on R^2? Explain your answer.
    (a) ((x_1,y_1),(x_2,y_2)) = x_1y_2 + x_2y_1

    (b) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + 7y_1y_2

    (c) ((x_1,y_1),(x_2,y_2)) = x_1x_2 + x_1y_2 + x_2y_1 + y_1y_2

    I know how to show its positive def. and I know how to show it is symmetric. I Want to know how to show its bilinear.

    Sorry I left out bilinear in my original question :/
     
  6. Mar 1, 2015 #5

    Stephen Tashi

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    What's the definition of "bilinear" in your course materials? Is it http://en.wikipedia.org/wiki/Bilinear_form ?
    Then show
    [itex] ( (u_x,u_y) + (v_x,v_y), (w_x,w_y) ) = ((u_x,u_y),(w_x,w_y)) + ((v_x,v_y)(w_x,w_y)) [/itex]
    and
    [itex]
    ( (u_x,u_y), (v_x,v_y)+ (w_x,w_y) ) = ((u_x,u_y),(v_x,v_y)) + ((u_x,u_y)(w_x,w_y)) [/itex]
    and
    [itex] (\lambda(u_x,u_y), (v_x,v_y)) = ((u_x,u_y),\lambda(v_x,v_y)) = \lambda( (u_x,u_y),(v_x,v_y)) [/itex]
     
  7. Mar 1, 2015 #6
    Oh right okay. Thanks :smile:
     
  8. Mar 3, 2015 #7

    Ray Vickson

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    Also, most (all?) discussions of "inner product" would include the condition ##((x_1,y_1),(x_1,y_1)) \geq 0##, with ##>0## holding whenever ##(x_1,y_1) \neq (0,0)##. I don't know if your textbook or notes includes this, but if it does then you need to check that as well.
     
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