Definition of homogeneity

Question

My book defines a system of linear equations to be homogeneous if the constant term in each equation is zero. And then it says if [A|0] (where A is the coefficient matrix) is a homogeneous system of m linear equations with n variables, where m is less than n then the system has infinitely many solutions. It also says that if m greater than or equal to n, the system has either a unique solution or infinitely many solutions. Why is this?

Answer 1

The system in vector notation is: $$ A \vec x=\vec 0 $$

so it has the the trivial solution $\vec x = \vec 0$. Suppose that there is another solution $\vec x_1\ne \vec 0$ such that $Ax_1=0$ then we have: $$ A(k \vec x_1)=k(A \vec x_1)=0 $$ so, also all the ( infinitely many) vectors $kx_1$ with $k \in \mathbb{R}$ ( or the field of the given vector field) are solutions.

And, if we have $\vec x_1,\vec x_2,...\vec x_m$ solutions, you can easily see, using linearity, that any liner combination of such vectors is a solution.