Results 1 to 2 of 2

Thread: Some Linear Algebra

  1. #1
    Junior Member
    Joined
    Sep 2008
    From
    Boston, Massachusetts
    Posts
    41

    Some Linear Algebra

    For the following, show that $\displaystyle M_{m\times n}(\mathbb{R}) $ does not form a vector space over $\displaystyle \mathbb{R} $ under the (non standard) operations $\displaystyle \hat{+}$ (addition) and $\displaystyle *$ (multiplication).

    (a) $\displaystyle A_{ij} \ \hat{+} \ B_{ij} = A_{ij} + B_{ij} $
    $\displaystyle k * A_{ij} = k^2A_{ij} $

    (b) $\displaystyle A_{ij} \ \hat{+} \ B_{ij} = A_{ij}B_{ij} $
    $\displaystyle k * A_{ij} = {A^k}_{ij} $
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by funnyinga View Post
    For the following, show that $\displaystyle M_{m\times n}(\mathbb{R}) $ does not form a vector space over $\displaystyle \mathbb{R} $ under the (non standard) operations $\displaystyle \hat{+}$ (addition) and $\displaystyle *$ (multiplication).

    (a) $\displaystyle A_{ij} \ \hat{+} \ B_{ij} = A_{ij} + B_{ij} $
    $\displaystyle k * A_{ij} = k^2A_{ij} $
    multiplication by scalar is obviously not linear.

    (b) $\displaystyle A_{ij} \ \hat{+} \ B_{ij} = A_{ij}B_{ij} $
    $\displaystyle k * A_{ij} = {A^k}_{ij} $
    the additive inverse doesn't always exist! see that the additive identity, namely 0, here is the matrix with all entries equal 1. now let $\displaystyle A$ be the matrix with 0 in the first row and first column and

    whatever you want everywhere else. then $\displaystyle A$ has no additive inverse.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Aug 1st 2011, 10:00 PM
  2. Basic Linear Algebra - Linear Transformations Help
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Dec 7th 2010, 03:59 PM
  3. Replies: 2
    Last Post: Dec 6th 2010, 03:03 PM
  4. Replies: 7
    Last Post: Aug 30th 2009, 10:03 AM
  5. Replies: 3
    Last Post: Jun 2nd 2007, 10:08 AM

Search Tags


/mathhelpforum @mathhelpforum