Results 1 to 4 of 4
Like Tree1Thanks
  • 1 Post By MathoMan

Math Help - Proving (u X v).(x X y) and using particular standard basis

  1. #1
    Senior Member x3bnm's Avatar
    Joined
    Nov 2009
    Posts
    300
    Thanks
    16

    Proving (u X v).(x X y) and using particular standard basis

    On page 13 of doCarmo's book on Differential geometry of curves and surfaces he stated that:

    ...we prove the relation:
    (u \wedge v)\cdot(x \wedge y) = \left | \begin{array}{cc} u\cdot x & v\cdot x \\ u \cdot y & v \cdot y \end{array} \right |

    where all u,v,x,y are arbitrary vectors. This can easily be done by observing that both sides are linear in u,v,x,y Thus it suffices to check that:
    (e_i \wedge e_j)\cdot (e_k \wedge e_l) = \left | \begin{array}{cc} e_i\cdot e_k & e_j\cdot e_k \\ e_i \cdot e_l & e_j \cdot e_l \end{array} \right |

    for all i,j,k,l = 1, 2, 3



    My question: what does it mean when the author said that

    " \text{... can easily be done by observing that both sides are linear}"? What did he mean by the word linear?

    Also the author said that:

    \text{Thus it suffices to check that:}

    (e_i \wedge e_j)\cdot (e_k \wedge e_l) = \left | \begin{array}{cc} e_i\cdot e_k & e_j\cdot e_k \\ e_i \cdot  e_l & e_j \cdot e_l \end{array} \right |

    where he defined e_1 \text{ as } (1,0,0)\,\, e_2 \text{ as } (0,1,0) \text{ and }\,\, e_3 \text{ as } (0,0,1)?

    Why is it sufficient to prove the above statement of the author for standard basis (1,0,0), (0,1,0), (0,0,1) only?

    How does this proof using standard basis automatically validates for non-standard basis? I know the author is right. But isn't it something like choosing a particular basis to prove our theorem which can be false for other non-standard bases?

    I can't find the answer. Is it possible to kindly help me detect what I'm lacking in understanding this statement?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Sep 2010
    Posts
    185
    Thanks
    13

    Re: Proving (u X v).(x X y) and using particular standard basis

    All bases are equivalent! So prove something in one basis and you have proven it for all possible bases. There is a similarity matrix that can be applied to transform everything from one basis to another.
    Thanks from x3bnm
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member x3bnm's Avatar
    Joined
    Nov 2009
    Posts
    300
    Thanks
    16

    Re: Proving (u X v).(x X y) and using particular standard basis

    Quote Originally Posted by MathoMan View Post
    All bases are equivalent! So prove something in one basis and you have proven it for all possible bases. There is a similarity matrix that can be applied to transform everything from one basis to another.
    Thanks a lot.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Sep 2010
    Posts
    185
    Thanks
    13

    Re: Proving (u X v).(x X y) and using particular standard basis

    Quote Originally Posted by x3bnm View Post
    Thanks a lot.
    Glad I could help.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] M 2x2 matrices of C with standard basis.
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 30th 2011, 01:25 AM
  2. Replies: 2
    Last Post: March 10th 2011, 02:39 PM
  3. standard basis
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 19th 2010, 08:12 AM
  4. Standard Matrices and basis vectors
    Posted in the Advanced Algebra Forum
    Replies: 10
    Last Post: June 23rd 2010, 09:28 PM
  5. Matrix with respect to the standard basis
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 26th 2009, 02:31 PM

Search Tags


/mathhelpforum @mathhelpforum