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

  • Thread starter Maxwhale
  • Start date
  • #1
35
0

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

and I am lost
 

Answers and Replies

  • #2
329
1
Let's go back to basics. What is the definition of Column space?
 
  • #3
35
0
the subspace of Rn spanned by the column vectors of A
 
  • #4
329
1
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.
 

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

  • Last Post
Replies
2
Views
970
Replies
2
Views
765
Replies
1
Views
1K
  • Last Post
Replies
5
Views
1K
Replies
1
Views
1K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
5
Views
9K
Replies
3
Views
2K
  • Last Post
Replies
12
Views
6K
  • Last Post
Replies
11
Views
11K
Top