a)if A is 6x8 what is the smallest possible dimension of N(A) ? the largest possible dim of N(A) b) if A is 6x4 what is smallest possible dim of N(A) ? the largest possible dim of R(A) ? smallest possible dim of N(At )? c) if A is 4x7 and rank (A) =3 what is dim N(A) ? does R(A) = R^3? why or why not? Solution a) There are 6 rows, and 8 columns, so each row could be independent. If so, they could all map to independent vectors, so the minimal dimension is 8-6=2. All vectors could map to 0, so the maximum dimension is 8. b) Since there are 6 rows.. if they are all independent, N(A)=0. The range space can be at most dim 4. For N(At) the minimum dimension is 2 (as in part a) c) We know that dim = rank + null, so 7 = 3 + N(A) so N(A)=4 R(A) is not R3... R(A) lies in R7 and is a subspace of that. There is, however, a bijection between R(A) and R3..