Results 1 to 2 of 2

Thread: Linear Transformations: im(S+T) subset of im(S) + im(T)

  1. #1
    Newbie
    Joined
    Mar 2011
    Posts
    1

    Linear Transformations: im(S+T) subset of im(S) + im(T)

    Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.

    (i) Show that im(S+T) is a subset of im(S) + im(T)
    (ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    22
    Quote Originally Posted by csmajor9 View Post
    Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V.

    (i) Show that im(S+T) is a subset of im(S) + im(T)
    (ii) Show that r(ST) <= min(r(S),r(T)), and that n(ST) <= n(S) + n(T)
    I'll help with i) and the first part of ii) and leave the rest to you. For i) merely note that if $\displaystyle v\in\text{im}\left(S+T\right)$ then $\displaystyle v=(S+T)(w)$ for some $\displaystyle w\in V$. But, $\displaystyle (S+T)(w)=S(w)+T(w)\in \text{im}(S)+\text{im}(T)$...conclude.

    For the first part of ii) the fact that $\displaystyle \text{rk}(ST)\leqslant \text{rk}(S)$ is clear since $\displaystyle \text{im}(ST)=(ST)(V)=S(T(V))\subseteq S(V)=\text{im}(S)$. Now, to see that $\displaystyle \text{rk}(ST)\leqslant \text{rk}(T)$ note that in general the image of a linear homomorphism has lesser dimension than the domain etc.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Linear Transformations and the General Linear Group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 26th 2011, 10:50 AM
  2. Basic Linear Algebra - Linear Transformations Help
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Dec 7th 2010, 03:59 PM
  3. Linear Algebra: Linear Transformations(1)
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Dec 25th 2008, 01:21 AM
  4. Subset, Subspaces and Linear Algebra
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Dec 18th 2008, 08:36 PM
  5. Replies: 3
    Last Post: Jun 2nd 2007, 10:08 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum