Results 1 to 3 of 3

Math Help - Linear maps and vectors.

  1. #1
    Member
    Joined
    May 2008
    Posts
    186

    Linear maps and vectors.

    A linear map L: R^n -> R is linear for all v, w that exist in R^n and all a, b that exist in R if...

    L(av + bw) = aL(v) + bL(w)

    Show that for any vector u that exists in R^n, the transposed vector u^T represents a linear map of type R^n -> R

    Also, prove conversely that a linear map L:R^n -> R can be represented as the transpose u^T of a vector u that exists in R^n.

    Sorry for the lack of latex usage. I don't know all the symbols and I'm hoping for an answer as soon as possible.

    Thanks for your help.

    Sean.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by sean.1986 View Post
    A linear map L: R^n -> R is linear for all v, w that exist in R^n and all a, b that exist in R if...

    L(av + bw) = aL(v) + bL(w)

    Show that for any vector u that exists in R^n, the transposed vector u^T represents a linear map of type R^n -> R
    define f: \mathbb{R}^n \longrightarrow \mathbb{R} by f(v)=u^T v.


    Also, prove conversely that a linear map L:R^n -> R can be represented as the transpose u^T of a vector u that exists in R^n.

    Sorry for the lack of latex usage. I don't know all the symbols and I'm hoping for an answer as soon as possible.

    Thanks for your help.

    Sean.
    suppose f: \mathbb{R}^n \longrightarrow \mathbb{R} is a linear map. let e_j \in \mathbb{R}^n be a vector with 1 in the jth row and 0 everywhere else. note that every v \in \mathbb{R}^n can be written (uniquely) as v=\sum_{j=1}^n v_je_j, where v_j \in \mathbb{R}.

    let u= \sum_{j=1}^n f(e_j)e_j. then since f is \mathbb{R}-linear, we have: f(v)=f \left(\sum_{j=1}^n v_je_j \right)=\sum_{j=1}^n v_jf(e_j)=\begin{bmatrix} f(e_1) & . & . & . & f(e_n) \end{bmatrix} \begin{bmatrix} v_1 \\ . \\ . \\ . \\ v_n \end{bmatrix} = u^T v.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    186
    Thanks but I'm still having some trouble understanding this. Is there any way of explaining it in simpler terms?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 10th 2010, 12:38 PM
  2. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: October 9th 2010, 04:42 AM
  3. Linear Maps
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 15th 2009, 10:27 AM
  4. Linear maps
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 27th 2008, 04:25 AM
  5. Linear maps. Proving its linear, and describing kernal.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 20th 2008, 01:46 AM

Search Tags


/mathhelpforum @mathhelpforum