How to check linear algebra work? Is there any way I can check whether I'm getting the right answers in linear algebra? In analysis I can just check maxima, minima, sketch the graph etc. to find out if I'm doing things right but in Linear algebra I'm not so sure?
I'm asked to find the equation of the plane of all points equidistant to $(4,3,-2)$ and $(7,-2,4)$  and I've got $-3x+5y-6z = -8$ but how can I be sure?
 A: For this case, you can pick a few points in the plane, and calculate the distance to the two points you mentioned using the formula $d = \sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2}$.  (To get points in the plane, just choose values for two of the coordinates, and use the plane equation to get the third.  For example, if I choose $x=1$ and $y=-1$, then $z=0$ gives a point in the plane.)
If you get three non-colinear points that satisfy the plane's equation, and the distances from those points to the two reference points match up, then I think you're safe in assuming that you're correct, because the equation you have is in the correct form for a plane.
A: Calculate the distances. Let the plane be $S$. If $(x,y,z)\in S$ then the difference of squares of distances from $(x,y,z)$ to those points is
$(x-4)^2 + (y-3)^2 + (z+2)^2 - ((x-7)^2+(y+2)^2 + (z-4)^2) =6 x-10 y+12 z-40$
Now this should be zero, so unless I made an error, I think you should have $-20$ in the equation of the plane instead of $-8$.
