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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.
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