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SIZIF [17.4K]
3 years ago
14

let Ax=b be any consistent system of linear equations, and let x1 be a fixed solution. show that every solution to the system ca

n be written in the form x=x1+x0, where x0 is a solution to Ax=0. show that every matrix of this form is a solution.
Mathematics
1 answer:
sweet [91]3 years ago
6 0
<span>There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
 

Ax=b and Ax+0
 
Specifically, if x1 is any specific solution to the linear system Ax = b, then the entire solution set can be described as
     
x1 + x0 : x0 is any solution to Ax=0
 
Geometrically, this says that the solution set for Ax = b is a translation of the solution set for Ax = 0. Specifically, the flat for the first system can be obtained by translating the linear subspace for the homogeneous system by the vector x1.
 
This reasoning only applies if the system Ax = b has at least one solution. This occurs if and only if the vector b lies in the image of the linear transformation A.</span>
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