Chapter 4 - Vector Spaces - 4.6 Rank of a Matrix and Systems of Linear Equations - 4.6 Exercises - Page 199: 8

(a) A basis for the row space is given by $$S=\{(1,-3,2),(0,35,7)\}.$$ (b) The rank of the matrix is $2$.

Let $A$ be given by $$ A= \left[\begin{array}{ccc}{2} & {-3} & {1} \\ {5} & {10} & {6} \\ {8} & {-7} & {5}\end{array}\right] $$ The reduced form of $A$ is given by $$\left[ \begin {array}{ccc} 1&-3&2\\ 0&{35}&7\\ 0&0&0\end {array} \right] $$ (a) A basis for the row space is given by $$S=\{(1,-3,2),(0,35,7)\}.$$ (b) The rank of the matrix is $2$.