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Suppose that the solutions of an equation $A \mathbf{x}=\mathbf{b}$ are all of the form $\mathbf{x}=x_{3} \mathbf{u}+\mathbf{p},$ where $\mathbf{u}=\left[\begin{array}{r}{5} \\ {1} \\ {-2}\end{array}\right]$ and $\mathbf{p}=\left[\begin{array}{r}{1} \\ {-3} \\ {4}\end{array}\right]$ Find points $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ such that the solution set of $A \mathbf{x}=\mathbf{b}$ is aff $\left\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\}$

$\begin{aligned} v_{1} &=\left[\begin{array}{c}{1} \\ {-3} \\ {4}\end{array}\right] \\ v_{2} &=\left[\begin{array}{c}{6} \\ {-2} \\ {2}\end{array}\right] \end{aligned}$

Calculus 3

Chapter 8

The Geometry of Vector Spaces

Section 1

Affine Combinations

Vectors

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