Let's say I have a system of linear equations with a free variable which has a solution set that looks like this:
$$\left\{\begin{pmatrix}x\\y\\z\end{pmatrix}=
\begin{pmatrix}\frac{11}{5}\\\frac{3}{5}\\0\end{pmatrix}+
\begin{pmatrix}-\frac{3}{5}\\\frac{1}{5}\\1\end{pmatrix}t\right\}
$$
with $t\in\mathbb{R}$.
My question is if I can "simplify" the vectors in the solution set in any way, by multiplying the vectors horizontally or vertically to remove the denominator $5$. Is it possible? Can I only multiply by $5$ the vector which is muliplied by the parameter?
I know that given an invertible square matrix $A$ and the system $A\vec{x}=\vec{b}$, there exists a unique solution $x=A^{-1}\vec{b})$. Can this help me with this question?
I think I've seen textbooks simplify when it comes to the basis of a subspace, but I'm not sure about solution sets with a parameter.
Best Answer
$$\begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} 1\\1\\2 \end{pmatrix} + \begin{pmatrix} -3\\1\\5 \end{pmatrix} s$$ where $s=(t-2)/5$.