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Prove the parallelogram law

  1. Jan 7, 2004 #1
    Hello everyone! =)

    This might be a good challenge to everyone here... =)

    1.) Let A be an n X n matrix and let x and y be vectors in R^n.
    Show that Ax .y = x.(A^T)(y), where "." means dot product and T is 'transpose'.

    2. show that u.v = (1/4)// u + v //^2 - (1/4)//u - v//^2
    where u and v are vectors; "." means dot product and
    //.....// denote the length of a vector.

    3. Prove the parallelogram law: // u + v //^2 + // u - v //^2 =
    2 //u//^2 + 2 //v//^2.

    4. Prove the Vandermonde determinant.
     
    Last edited: Jan 10, 2004
  2. jcsd
  3. Jan 7, 2004 #2
    [tex](\vec{u}+\vec{v})^2=\vec{u}^2+\vec{v}^2+2\vec{u}.\vec{v}[/tex]

    [tex](\vec{u}-\vec{v})^2=\vec{u}^2+\vec{v}^2-2\vec{u}.\vec{v}[/tex]

    [tex](\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2 = 4\vec{u}.\vec{v}[/tex]

    [tex]\frac{(\vec{u}+\vec{v})^2 - (\vec{u}-\vec{v})^2}{4} = \vec{u}.\vec{v}[/tex]
     
    Last edited: Jan 7, 2004
  4. Jan 13, 2004 #3
    Hello there!

    Hi.

    You're right. Well, if you pretty know the property involved on the right side, you could expand the following...

    //u + v//^2 = //u//^2 + //v//^2 + 2 (u.v) and
    //u - v//^2 = //u//^2 + //v//^2 - 2 (u.v) Thus

    u.v = (1/4)//u//^2 + (1/4)//v//^2 + 1/2(u.v) - (1/4)//u//^2 - (1/4)//v//^2 + 1/2(u.v)

    u.v = 1/2(u.v) + 1/2(u.v)
    = u.v
     
  5. Jan 15, 2004 #4
    Parallelogram Law

    In #3, it is very easy, as long as you know the equivalent of
    //u + v//^2 and //u - v//^2 (look on the previous replies.)

    the sum of the two yields:

    //u//^2 + //v//^2 + 2(u.v) + //u//^2 + //v//^2 - 2(u.v)

    and thus = 2//u//^2 + 2//v//^2

    In Vandermonde determinant, use the cofactor for easier proving.
     
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