Chapter 2 - Matrix Algebra - 2.8 Exercises - Page 153: 13

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Any vector x such that $x=\begin{bmatrix} 1\\ -2\\ 1\\ 0 \end{bmatrix}x_3+\begin{bmatrix} -1\\ 4\\ 0\\ 1 \end{bmatrix}x_4 $ where $x_3$ and $x_4$ are free variables will be in Nul A. One example is $\begin{bmatrix} 0\\ 2\\ 1\\ 1 \end{bmatrix} $ Any x such that $x=\begin{bmatrix} 3\\ -9\\ 9\\ \end{bmatrix}c_1+\begin{bmatrix} 2\\ -4\\ 2\\ \end{bmatrix}c_2+\begin{bmatrix} 1\\ 1\\ -5\\ \end{bmatrix}c_3+\begin{bmatrix} -5\\ 7\\ 1\\ \end{bmatrix}c_4 $ where $c_1,c_2,c_3,c_4$ are scalars is in Col A. One example is $\begin{bmatrix} 1\\ 1\\ -5\\ \end{bmatrix}$