Chapter 4 - Vector Spaces - 4.6 Rank of a Matrix and Systems of Linear Equations - 4.6 Exercises - Page 199: 5

(a) $$S=\{(1,0), (0,1)\}.$$ (b) The rank of the matrix is $2$.

(a) Since the reduced row-echelon form of the matrix is $\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$, then the basis for the row space is $$S=\{(1,0), (0,1)\}.$$ (b) Since the dimension of the row space is $2$, then the rank of the matrix is $2$.