1. Not finding help here? Sign up for a free 30min 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!

A problem on the Image Space

  1. Nov 15, 2015 #1
    1. The problem statement, all variables and given/known data

    If A is an mxn matrix, show that for each invertible nxn matrix V, im(A) = im(AV)

    2. Relevant equations
    none

    3. The attempt at a solution
    I know that im(A) can also be written as the span of columns of A.
    I also know that AV = [Av1 Av2 ... Avn]
    so im(AV) is the span of the columns of that matrix. However, I don't understand how the two can be equal.
     
  2. jcsd
  3. Nov 15, 2015 #2

    fresh_42

    Staff: Mentor

    Forget the spanning vectors for a moment. What does it mean for a vector x to be in ##im A##. ##im (A V)## resp.?
     
  4. Nov 15, 2015 #3
    If A is mxn and y ∈ im(A), then y can be written as Ax, where x ∈ Rn.
    If y ∈ im(AV) then y can be written as (AV)x, where x ∈ Rn.
     
  5. Nov 15, 2015 #4

    fresh_42

    Staff: Mentor

    Right. Now all you need is the associative law for linear functions for one inclusion and to put ##V \cdot V^{-1} = 1## somewhere in between for the other inclusion. ##im (A \cdot V) ⊆ im A## and ##im (A \cdot V) ⊇ im A##

    Actually you've already proved one inclusion by explaining to me.
     
  6. Nov 15, 2015 #5
    I believe I understand it! Could you please check if I've got it right?

    Assume y ∈ im A
    then y = Ax = (AVV-1)x
    y = AV(V-1x)
    since V-1x ∈ Rn, then y ∈ im(AV) and im(A) ⊆ im(AV)

    Assume y ∈ im AV
    then y = AVx = A(Vx)
    since Vx ∈ Rn, then y ∈ im(A) and im(AV) ⊆ im(A)

    Therefore im(A) = im(AV).
     
  7. Nov 15, 2015 #6

    fresh_42

    Staff: Mentor

  8. Nov 15, 2015 #7
    Thanks a lot!! I really appreciate it :)
     
  9. Nov 15, 2015 #8

    fresh_42

    Staff: Mentor

    You're welcome.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: A problem on the Image Space
  1. Image problem (Replies: 3)

Loading...