My question is with respect to the solution to PS2-part 2-2c (will attach as screenshot. why does one row’s values being the average of the correspoinding values in the other two rows of a matrix imply, in general that the rows are coplanar?

Question

anonymous · Answer

This is a screenshot of the problem.

phi · Answer

If you have two vectors that are not parallel, then they define a plane. (That means any point on the plane can be "reached" by some linear combination of the two vectors)

the 3 rows (or 3 columns ) of the matrix represent 3 different vectors.

 if one of the vectors is a linear combination of the other 2, then that 3rd vector is not "independent", which is to say, using the 3rd vector does not allow you to reach any point that can't be reached using the first 2 vectors.  And as noted, the two vectors define a plane, so the third vector must lie in this plane.

phi · Answer

These ideas are covered more deeply in 18.06 example http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-9-independence-basis-and-dimension/

anonymous · Answer

Thanks again