If w is a subspace of r", then the set of all vectors in rn that are orthogonal to every vector in w is called the orthogonal complement of w, and is denoted by the symbol w. that is wi-lu e r" : u. tv = 0 for all zu e w} 3. (5 points) if a is an mxn matrix, prove that the null space of a and the row spaoe of a are orthogonal complements in r". for example, show that n(a) (note that r(a) = (n(a))" is also true) (r(a)) 4. (5 points) if a is an m x n matrix, prove that the null space of at and the column space of a are orthogonal complements in r for example, show that n(a") = (c(a))". (note that c((a"))" is also true) note: because of theorem 4.8.6 (c) see text page 253, that is (w1)" = w once we have n(a) (r( then r(a) (n(a)) will also follows. simi- larly, once we have n((a))", then c(a) = (n(a"))" will also follows. note: the results obtained in problem 3 and 4 are also refered to as funda- mental theorem of linear algebra, this shows its important in the subject.