dont you think that is a bit oversimplified, doctor?
actually squeaks question would best be asnwered by a method of solving a set of linear equations that me and some of the boys at MIT have been working on:
A system of linear equations is homogeneous if all of the constant terms are zero:
begin{alignat}{7} a_{11} x_1 &&; + ;&& a_{12} x_2 &&; + cdots + ;&& a_{1n} x_n &&; = ;&&& 0 \ a_{21} x_1 &&; + ;&& a_{22} x_2 &&; + cdots + ;&& a_{2n} x_n &&; = ;&&& 0 \ vdots;;; && && vdots;;; && && vdots;;; && &&& ,vdots \ a_{m1} x_1 &&; + ;&& a_{m2} x_2 &&; + cdots + ;&& a_{mn} x_n &&; = ;&&& 0. \ end{alignat}
A homogeneous system is equivalent to a matrix equation of the form
Atextbf{x}=textbf{0}
[edit] 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.
[edit] Relation to nonhomogeneous systems
There is a close relationship between the solutions to a linear system and the solutions to the corresponding homogeneous system:
Atextbf{x}=textbf{b}qquad text{and}qquad Atextbf{x}=textbf{0}text{.}
Specifically, if p is any specific solution to the linear system Ax = b, then the entire solution set can be described as
left{ textbf{p}+textbf{v} : textbf{v}text{ is any solution to }Atextbf{x}=textbf{0} right}.
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 p.
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.