I've been reading Linear Algebra by Jacob for self study and I'm wondering about the geometric interpretation of a unique solution to systems of equations in $\mathbb R^3$.
For example, the homogeneous system $$
\begin{eqnarray*}
x+5y-2z &=&0, \\
x-3y+z &=& 0, \\
x+5y-z &=& 0
\end{eqnarray*}
$$ has the unique solution $x=y=z=0$. From Calc III, I would think of this as being 3 planes and the various $(x,y,z)$ values of the line (vector) that forms the intersection as the solution to the system. But the solution here and in many other systems in $\mathbb R^3$ is a point and that doesn't seem to be possible.
Am I wrong in trying to understand this geometrically or is there something I'm missing?
Sorry if this is hard to read, I haven't learning latex yet and thanks to everyone who responded to my question about self study books for real analysis.
Best Answer
You're correct in your interpretation it is the intersection of three planes. Three planes can indeed intersect at one point for example, the origin is the only point contained in $x = 0, y = 0, z = 0$.