Use theorem 4.2.1 to determine which of the following are subspaces of mnn. (a) the set of all diagonal n × n matrices. (b) the set of all n × n matrices a such that det(a) = 0. (c) the set of all n × n matrices a such that tr(a) = 0. (d) the set of all symmetric n × n matrices. (e) the set of all n × n matrices a such that at = −a. (f) the set of all n × n matrices a for which ax = 0 has only the trivial solution. (g) the set of all n × n matrices a such that ab = ba for some fixed n × n matrix b

Write a real-world word problem that can be solved by adding 34.99 and 5.79. then solve

Here is their argument. given the obtuse angle x, we make a quadrilateral abcd with ∠dab = x, and ∠abc = 90◦, and ad = bc. say the perpendicular bisector to dc meets the perpendicular bisector to ab at p. then pa = pb and pc = pd. so the triangles pad and pbc have equal sides and are congruent. thus ∠pad = ∠pbc. but pab is isosceles, hence ∠pab = ∠pba. subtracting, gives x = ∠pad−∠pab = ∠pbc −∠pba = 90◦. this is a preposterous conclusion – just where is the mistake in the "proof" and why does the argument break down there?

How can you check that you have correctly converted frequencies to relative frequencies?