Solving an Inhomogeneous System: A Coset of W in R^m

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In summary, the solution of an inhomogeneous system AX=B forms a coset of W, where W is the additive subgroup of R^m of solutions of a system of homogeneous equations AX=0. This can be shown by considering a solution T of AX=B and showing that W+T is the set of solutions of AX=B, thus forming a coset. The definition of a coset is a subset of the form a+H, where a is any element and H is a subgroup of a group G.
  • #1
hgj
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I need to do the following question:
Let W be the additive subgroup of R[tex]^m[/tex] of solutions of a system of homogeneous equations AX=0. Show that the solutions of an inhomogeneous system AX=B forms a coset of W.

I really just don't know where to start. Any help would be appreciated.
 
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  • #2
Okay, here's what I have now:
Let T be a solution of AX=B. Then W+T is the set of solutions of AX=B. So AT=B. Then AX=AT [tex]<=>[/tex] A(X-T)=0 [tex]<=>[/tex] X-T [tex]\in[/tex] W [tex]<=> [/tex] X [tex]\in[/tex] W+T

I think I don't fully understand the definition of coset, so I'm not sure what to do from here. Our definition is :
A left coset is a subbset of the form aH={ah s.t. h is in H} for any subgroup H of a group G.
 
  • #3
But this said 'Let W be the additive subgroup of Rm'!

Your cosets are defined by a+H={a+h s.t. h is in H}
 

1. What is an inhomogeneous system?

An inhomogeneous system is a system of linear equations where the right-hand side contains a non-zero constant term. This means that the equations are not equal to zero and do not have a trivial solution.

2. What is a coset?

A coset is a subset of a group that contains all elements of the group combined with a specific element. In linear algebra, a coset of a subspace is a set of vectors that can be obtained by adding any vector from the subspace to a fixed vector.

3. How is a coset related to solving an inhomogeneous system?

In solving an inhomogeneous system, the coset of a subspace is used to find the particular solution to the system. This particular solution, when added to the solutions of the homogeneous system, gives the complete solution to the inhomogeneous system.

4. Can a coset be used to solve any inhomogeneous system?

Yes, a coset can be used to solve any inhomogeneous system as long as the system is linear and the subspace is a valid subspace of the vector space in which the system exists.

5. What are some applications of solving inhomogeneous systems using cosets?

Solving inhomogeneous systems using cosets has many applications in engineering, physics, and computer science. One common application is in solving systems of equations in circuit analysis. It is also used in image processing, signal processing, and machine learning algorithms.

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