Suppose A is a 13 times 9 matrix and the rref of A has 7 lea

Suppose A is a 13 times 9 matrix, and the rref of A has 7 leading entries Define T: R^9 rightarrow R^13 by 7(x) = Ax, and define S: R^13 rightarrow R^9 by S(x) = A^T x rank T= dim N(7) = rank S = dim N(S) = dim R(S) = dim R(7) = Give the maximum number of linearly independent rows in A.

Solution

dimension represents the number of free variables in rref

Column represents the number of non zero rows.

rank+nullity=dimension that is R(T)+N(T)=DIM T and R(S)+N(S)=DIM S

1) rank of T= 7

2) 4

3) 9

4) 4

5) 9

6) 5