(3) explain why t : r3 ? r3 given by t=
(x1)
(x2)
(x3) =
(x1x2)
(x2x3)
(x3x1)
is not a linear transformation by giving specific examples of x, y ? r3 such that t(x + y) 6= t(x)+ t(y) and examples of x ? r3, c ? r such that t(cx) 6= ct(x). (remark: only one property needs to fail to demonstrate t is not linear, but i have confidence you can find examples of both properties failing.)
(4) consider the linear transformation t : r2 ? r2 given by t
(x1)
(x2) =
(2x1 ? 3x2)
(cx1 + 3x2).
(a) write down the standard matrix a of t.
(b) for what value(s) of c does a have a nontrivial null space?
(5) answer all of the true/false questions in the exercises of sections 2.6, 3.1, and 3.2 of the textbook. explain how you know each is true or false. (for example, cite a definition or theorem if the statement is true, or provide a counterexample of the statement is false.)