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.