Chapter 1 - Linear Equations in Linear Algebra - Supplementary Exercises - Page 91: 19

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Parametric equation of a line in space through two points $v_{1}, v_{2}$ $x=v_{1}+t\left(v_{2}-v_{1}\right)$ Because $v_{3}$ is on that line, there is some $t_{1}$ so that $v_{3}=v_{1}+t_{1}\left(v_{2}-v_{1}\right)=v_{1}-t_{1} v_{1}+t_{1} v_{2}=\left(1-t_{1}\right) v_{1}+t_{1} v_{2},$ so $v_{3}$ is a linear combination of $v_{1}, v_{2}$