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

[Maxwhale]

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

 

[PowerIso]

Let's go back to basics. What is the definition of Column space?

 

[Maxwhale]

the subspace of Rn spanned by the column vectors of A

 

[PowerIso]

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.