# Chapter 4 - Vector Spaces - 4.6 Rank of a Matrix and Systems of Linear Equations - 4.6 Exercises - Page 199: 9

(a) A basis for the row space is given by $$S=\{(-2,-4,4,5),(0,0,0,1)\}.$$ (b) The rank of the matrix is $2$.

#### Work Step by Step

Let $A$ be given by $$A= \left[\begin{array}{rrrr}{-2} & {-4} & {4} & {5} \\ {3} & {6} & {-6} & {-4} \\ {-2} & {-4} & {4} & {9}\end{array}\right].$$ The reduced form of $A$ is given by $$\left[ \begin {array}{cccc} -2&-4&4&5\\ 0&0&0&1 \\ 0&0&0&0\end {array} \right]$$ (a) A basis for the row space is given by $$S=\{(-2,-4,4,5),(0,0,0,1)\}.$$ (b) The rank of the matrix is $2$.

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