G 4. (5 points) If A is an m × n matrix, prove that the null space of AT and the column space of A are orthogonal complements in R m. For example, show that N(AT ) = (C(A))⊥ . (Note that C(A) = N(AT ) ⊥ is also true). Note: Because of Theorem 4.8.6 (c) See Text Page 253, that is (W⊥) ⊥ = W, once we have N(A) = (R(A))⊥ , then R(A) = (N(A))⊥ will also follows. Similarly, once we have N(AT ) = (C(A))⊥ , then C(A) = N(AT ) ⊥ will also follows. Note: The results obtained in Problem 3 and 4 are also refered to as Fundamental Theorem of Linear Algebra, this shows its important in the subject

Ican interpret and represent the remainder in division calculations sensibly for the context (eg if we wont to know how many 40 seat buses will be needed to transport 170 students for on excursion, we divide 170 by 40 and realise buses and another smaller vehicle to that we need 4 transport 10 more students) solve the following questions. show your work! 11) the zoo had 420 butterflies that needed to be transported to a new enclosure. the boxes that the butterflies would be moved in only hold 80 butterflies in each. how many boxes would they need? 2 12) each classroom in the school can have exactly 25 students, if there are 345 students in the school how many full classrooms are there?