Our goal on this problem is to show that if A € R***, then depending upon A, it's possible
that the ker A is a proper subspace of the ker A², or ker(4) — ker (4²). This means the
ker A is smaller than ker A².
a. [This is the easy part] I'm certain all of you know that all squares are rectangles but
not all rectangles are squares. For this, give definitions of a rectangle and of a square.
From your definitions, why are all squares rectangles? How do you know not all
rectangles are squares? The latter involves coming up with a counterexample. If
you're not sure what a counterexample is, do an internet search on the term. Provide
the counterexample and show why, according to your definitions, it is a rectangle but
not a square.