Parametrization Homogeneous 39A237

1. **State the problem:** We want to find a parametrization for the solution set to the homogeneous equation $Ax=0$ using the given information about the solution set to $Ax=b$. 2. **Recall the general solution form:** The general solution to $Ax=b$ can be written as: $$x = x_p + x_h$$ where $x_p$ is a particular solution to $Ax=b$, and $x_h$ is the general solution to the homogeneous system $Ax=0$. 3. **Identify the particular and homogeneous parts from the given parametrization:** Given: $$\begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} = \begin{bmatrix} -6 \\ 4 \\ 1 \end{bmatrix} + x_2 \begin{bmatrix} -7 \\ 9 \\ 8 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 0 \\ 9 \end{bmatrix}$$ - The vector $\begin{bmatrix} -6 \\ 4 \\ 1 \end{bmatrix}$ is a particular solution $x_p$. - The vectors $\begin{bmatrix} -7 \\ 9 \\ 8 \end{bmatrix}$ and $\begin{bmatrix} 5 \\ 0 \\ 9 \end{bmatrix}$ span the null space (solutions to $Ax=0$). 4. **Write the parametrization for $Ax=0$:** The solution set to $Ax=0$ is all linear combinations of the null space vectors: $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} = x_2 \begin{bmatrix} -7 \\ 9 \\ 8 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 0 \\ 9 \end{bmatrix}$$ 5. **Final answer:** $$\boxed{\begin{bmatrix} x \\ y \\ z \end{bmatrix} = x_2 \begin{bmatrix} -7 \\ 9 \\ 8 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ 0 \\ 9 \end{bmatrix}}$$