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!

Solution of a Ax=b exists iff b is in CS (A) ?

  1. Nov 9, 2008 #1
    1. The problem statement, all variables and given/known data

    Show that Ax = b has a solution if and only if b is in CS(A).

    2. Relevant equations

    3. The attempt at a solution

    Ax = b
    b [tex]\in[/tex] CS(A) means
    d1A1 + d2A2+ .........+ dnAn = b

    and I am lost
  2. jcsd
  3. Nov 9, 2008 #2
    Let's go back to basics. What is the definition of Column space?
  4. Nov 9, 2008 #3
    the subspace of Rn spanned by the column vectors of A
  5. Nov 10, 2008 #4
    Yep, let's break this down into parts:

    1)Assume Ax = b has a solution, then b can be written as a linear combination with the vectors from the columns of A. Since the span is the linear combination of the vectors a1 a2 a3 ... an then b is in the span. So then use your definition.

    2)Now work the other way. Assume b is in Col A, and work towards showing that Ax = b has a solution because of that.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?