Answer
(a) $$S=\left\{(1,-3,2),(0,1,-\frac{1}{2})\right\}.$$
(b) The rank of the matrix is $2$.
Work Step by Step
(a)The reduced form of the matrix is given as follows
$$\left[\begin{array}{rrr}{1} & {-3} & {2} \\ {4} & {2} & {1}\end{array}\right] \rightarrow\left[\begin{array}{rrr}{1} & {-3} & {2} \\ {0} & {1} & {-\frac{1}{2}}\end{array}\right].$$
Hence, the basis for the row space is
$$S=\left\{(1,-3,2),(0,1,-\frac{1}{2})\right\}.$$
(b) Since the dimension of the row space is $2$, then the rank of the matrix is $2$.
