Question

let Ax=b be any consistent system of linear equations, and let x1 be a fixed solution. show that every solution to the system can be written in form x=x1+x2, where x0 is a solution to Ax=0. show that every matrix of this form is a solution.

Answer 1

I think there's a mistake in this question somewhere. Maybe you meant x = x0 + x1 + x2? or something similar.

Answer 2

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.

Answer 3

HOMOGENEOUS SYSTEM
A system of linear equations is homogeneous if all of the constant terms are zero:

a11x1 + a12x2 +...+ a1n Xn=0
a12x1 + a22 x2 +...+a2n xn=0
... .. .. ... ..
am1 x1 + am2 x2 +...+amn xn=0


A homogeneous system is equivalent to a matrix equation of the form

Ax=0

where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.
Solution set

Every homogeneous system has at least one solution, known as the zero solution (or trivial solution), which is obtained by assigning the value of zero to each of the variables. The solution set has the following additional properties:

1. If u and v are two vectors representing solutions to a homogeneous system, then the vector sum u + v is also a solution to the system.
2. If u is a vector representing a solution to a homogeneous system, and r is any scalar, then ru is also a solution to the system.

These are exactly the properties required for the solution set to be a linear subspace of Rn. In particular, the solution set to a homogeneous system is the same as the null space of the corresponding matrix A.

Answer 4

kind of got it.... so u said that Ax=b and Ax=0 are mostly similar and both are homogeneous.
so if Ax=0 can be express as x1-x2
any solution of Ax=b can be express as x1+x2 and also x1 +rx2 is a solution
I trying to decode..