Results 1 to 2 of 2

Thread: Norm of sum of vectors

  1. #1
    Senior Member Pinkk's Avatar
    Joined
    Mar 2009
    From
    Uptown Manhattan, NY, USA
    Posts
    419

    Norm of sum of vectors

    Given vectors $\displaystyle x_{1}, x_{2}, ..., x_{k}\in \mathbb{R}^{n}$, what is $\displaystyle |x_{1} + x_{2} + ... + x_{k}|^{2}$?

    So $\displaystyle |x_{1} + x_{2}|^{2} = |x_{1}|^{2} + |x_{2}|^{2} + 2x_{1}\cdot x_{2}$ and extending it to the sum of three vectors it seems to be $\displaystyle |x_{1}|^{2} + |x_{2}|^{2} + |x_{3}|^{2} + 2( x_{1}\cdot x_{2} + x_{1}\cdot x_{3} + x_{2}\cdot x_{3})$. So it seems to be that for the norm of $\displaystyle k$ vectors we have $\displaystyle |x_{1} + x_{2} + ... + x_{k}|^{2} = |x_{1}|^{2} + |x_{2}|^{2} + ... + |x_{k}|^{2} + 2(\sum x_{i}x_{j})$ for $\displaystyle i\ne j, 1\le i, j \le k$. Is this correct and is there a better way to formulate this? And how would I go about giving a formal proof for this problem? Thanks.

    Edit: I think I found a more simplified formulation: $\displaystyle \sum_{j=1}^{k}\sum_{i=1}^{k} x_{i}\cdot x_{j}$, but how would I go about giving a formal proof? Thanks.
    Last edited by Pinkk; Aug 30th 2010 at 05:28 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    7
    Awards
    2
    The norm of a vector can always be written as follows:

    $\displaystyle \|x\|=\sqrt{\langle x|x\rangle},$

    where $\displaystyle \langle x|y\rangle$ is the inner product of vector $\displaystyle x$ with vector $\displaystyle y$. The inner product is just a generalization of the dot product. Therefore,

    $\displaystyle \|x\|^{2}=\langle x|x\rangle.$ It follows that

    $\displaystyle \|x+y\|^{2}=\langle (x+y)|(x+y)\rangle=\langle x|x\rangle+\langle x|y\rangle+\langle y|x\rangle+\langle y|y\rangle.$

    I think this line of reasoning should enable you to see how to do your proof. How would you start?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Find vectors using the euclidean norm and inner product.
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Mar 9th 2011, 05:07 AM
  2. Replies: 3
    Last Post: Jul 13th 2010, 06:37 PM
  3. Norm of Vectors
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: Feb 22nd 2010, 02:11 AM
  4. Proof (vectors, norm, and dot product)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Nov 12th 2009, 08:58 AM
  5. Replies: 2
    Last Post: Nov 7th 2009, 12:13 PM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum