I'm just starting to learn linear algebra and something has been stumping me to no end, why does a row of zeroes, particularly in a $4\times 3$ matrix of linear systems, mean there's an infinite amount of solutions? I've read people saying it's because of the free variable which we can make whatever we want, so if we have $x=5t-4$ and $y=3t-5$ and $0=0$ (this being the third row of zeroes) then there's infinite solutions because $t$ can be whatever we want.
What I don't get is how if we looked at $t=2$ then $x$ and $y$ aren't equal. This may be my understanding in the geometry as I'm thinking of two lines on a graph, and that each intersection is a solution, which obviously these lines are different so they'd have one intersection. So basically what I'm asking is, what exactly is the free variable? Is the value of $x$ and $y$ important to the number of solutions? Should I stop thinking of these as lines intersecting on a graph? Lastly what should I be thinking about geometrically when I see this kind of matrix?
Best Answer
In a $4\times3$ matrix, every row corresponds to a plane, like
$$3x-y+2z=4.$$
If you choose two coordinates, the third is implied.
Three non-parallel planes intersect in a single point.
But the plane of equation
$$0x+0y+0z=0$$ is very special in that you can plug any coordinates and the equation is fulfilled. In fact, it doesn't define a plane but the whole space.
Then the intersection of two planes and the whole space is a straight line, with infinitely many points.
Similarly, with two zero rows, you get the intersection of a plane and the whole space, hence the same plane.