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

  • Thread starter Maxwhale
  • Start date
  • Tags
    Cs
In summary, the conversation discusses the relationship between the existence of a solution for Ax = b and the column space of matrix A. It is stated that the problem has a solution if and only if b is in the column space of A. The definition of column space is also discussed, which is the subspace of Rn spanned by the column vectors of A. Two potential approaches to proving the relationship are also mentioned.
  • #1
Maxwhale
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
 
Physics news on Phys.org
  • #2
Let's go back to basics. What is the definition of Column space?
 
  • #3
the subspace of Rn spanned by the column vectors of A
 
  • #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.
 

1. What does it mean for a solution to exist for Ax=b?

For a solution to exist for Ax=b, it means that there is a set of values for the variable x that satisfies the equation and makes it true. In other words, when we substitute these values into the equation, it will result in the given value of b.

2. What is the significance of CS (A) in the equation Ax=b?

CS (A) stands for the column space of the matrix A. It represents the span of all possible linear combinations of the columns in A. In the context of the equation Ax=b, it means that the value of b must be in the column space of A for a solution to exist.

3. How can we determine if b is in CS (A)?

We can determine if b is in CS (A) by checking if the columns of A are linearly independent. If they are linearly independent, then b is in the column space of A. We can also use methods such as row reduction or calculating the rank of A to determine if b is in CS (A).

4. Can a solution exist for Ax=b if b is not in CS (A)?

No, a solution cannot exist for Ax=b if b is not in CS (A). This is because the equation represents a system of linear equations, and the column space of A represents all possible solutions to the system. If b is not in CS (A), it means that there is no combination of the columns of A that can produce the value of b.

5. Are there any special cases where a solution may still exist if b is not in CS (A)?

Yes, there are special cases where a solution may still exist if b is not in CS (A). This can happen when the matrix A is not square, and there are more than one solution to the system of linear equations. In this case, b may not be in CS (A) but a solution to Ax=b still exists.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
648
  • Calculus and Beyond Homework Help
Replies
14
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
824
  • Calculus and Beyond Homework Help
Replies
3
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
674
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
2
Replies
62
Views
7K
  • Calculus and Beyond Homework Help
Replies
3
Views
965
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Back
Top