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

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Homework Statement



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

Homework Equations





The Attempt at a Solution



Ax = b
b \in CS(A) means
d1A1 + d2A2+ ...+ dnAn = b

and I am lost
 
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Let's go back to basics. What is the definition of Column space?
 
the subspace of Rn spanned by the column vectors of A
 
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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