Properties of systems of linear equations

[hholzer]

If you have, as an example, a 3x3 matrix:

A = { {1, 2, 3}, {4, 5, 6}, {7, 8, 9} }

Such that Ax = 0

and you perform row reductions on it
and at a certain point it reduces to:
{ {1, 2, 3}, { 0, 1, 2}, {0, 1, 2} }

Then you could legitimately remove
one of {0, 1, 2} from the system
as it contributes nothing special, right?

As such, this would indicate that you have
an infinite number of solutions with a
free variable, yes?

(Geometrically: if in the 3x3 case above
you end up with two rows that are the
same after row reductions, then it would
suggest that having a single point as a
solution is an impossibility. As a single point
could only arise from three separate planes.).

 

[HallsofIvy]

What "system" are you talking about removing "{0, 1, 2}" from? You said nothing about a system before that.

If you are thinking about that matrix as representing a system of equations, say x+ 2y+ 3z= a, 4x+ 5y+ 6z= b, and 7x+ 8y+ 9z= c, then the augmented matrix is
$$\begin{bmatrix}1 & 2 & 3 & a \\ 4 & 5 & 6 & b \\ 7 & 8 & 9 & c\end{bmatrix}$$

Subtracting 4 times the first row from the second and 7 times the first row from the third gives
$$\begin{bmatrix} 1 & 2 & 3 & a \\0 & -3 & -6 & b- 4a \\ 0 & -6 & -12 & c- 7a\end{bmatrix}$$
Dividing the second row by -3 and the third row by -6 gives
$$\begin{bmatrix} 1 & 2 & 3 & a \\ 0 & 1 & 2 & \frac{4a- b}{3} \\ 0 & 1 & 2 & \frac{7a- c}{6}\end{bmatrix}$$
In which the first three columns are now the same as yours. Subtracting the second row from the third gives
$$\begin{bmatrix} 1 & 2 & 3 & a \\ 0 & 1 & 2 & \frac{4a- b}{3} \\ 0 & 0 & 0 & \frac{2b- a- c}{6}\end{bmatrix}$$.

But we cannot simply ignore that last row. It corresponds to the equation
$$0x+ 0y+ 0z= \frac{2b- a - c}{6}$$
which is true only if 2b- a- c= 0. If that is true, then there exist an infinite number of solutions to the original system. If it is not true, then there is NO solution.

(Geometrically, if two or all three of the planes are parallel then there is NO solution.)

 

[hholzer]

I had made mention of Ax = 0 because I was talking about
the homogeneous case.