1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)

  1. Apr 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)


    2. Relevant equations



    3. The attempt at a solution

    I think I need to show AB is a linear combination of the columns of A, but I'm not sure how to show this in general.
     
  2. jcsd
  3. Apr 16, 2013 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    welcome to pf!

    hi jazz_hole! welcome to pf! :smile:
    isn't it obvious? :wink:
     
  4. Apr 16, 2013 #3
    I can see how it is true, but I don't know how to show it rigorously. For all n x n matrices.
     
  5. Apr 16, 2013 #4

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Assume ##y \in \text{range}(AB)##. Show that ##y \in \text{range}(A)##.

    You need to understand what is meant precisely when you say ##y \in \text{range}(AB)##.
     
  6. Apr 16, 2013 #5

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    start "let y ε range (AB)

    then there exists …" :wink:
     
  7. Apr 16, 2013 #6

    berkeman

    User Avatar

    Staff: Mentor

    Moderator note -- The OP has been banned because of his choice of username.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Suppose A and B are n × n matrices. Show that range(AB) ⊆ range(A)
  1. Lim a(n)b(n) = AB (Replies: 2)

Loading...