Can changing representation reveal a solution?
A different representation makes certain relationships directly visible. Compare which calculation each representation makes easier and which information it preserves.
Read on through a shared source
How can columns explain a matrix equation? →Translate coordinates into a geometric picture. The columns of a matrix then show which combinations of vectors can reach the desired output.Which equations survive a collapse in dimension? →Once a transformation flattens space, reachable outputs lie in its image. A target inside that image and one outside it lead to different answers.What is the same in a diagram and a column calculation? →Switch between geometric motion and coordinate arithmetic to see one linear operation from two angles. Strang’s column combination is a worked example; neither representation is best for every task.
Compare the passages
choose the appropriate
data structure before you start tinkering
around with the algorithm.that right-hand
side that I'm trying to get is actually one
of these columns.