Question

When a problem says to solve a system using gaussian elimination and express the solution in "vector form", how do you express the result in vector form?

Answer 1

I created an augmented matrix, and got it to reduced row echelon form. Now how do I express that in vector notation?

Answer 2

\[\left[\begin{matrix}1 & 0 & 0 &-5 \\ 0 & 1 & 0 &7\\ 0 & 0 & 1 &-9\end{matrix}\right]\]

Answer 3

The solution vector is the last column of that matrix. The augmented matrix you've written is shorthand for
\[\left[\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right] \cdot \left[\begin{matrix}x \\y \\ z\end{matrix}\right] =\left[\begin{matrix}-5 \\7\\ 9\end{matrix}\right] \]

Answer 4

Thats what is confusing me. Every time I have expressed a solution as a vector it has parameters, because there is at least one free variable. In this case there are no free variables and the RREF produces a solution that is geometrically interpreted as the point at which three planes intersect. So is the vector representation.....\[(x,y,z)^T =\left(\begin{matrix}-5 \\ 7\\ 9\end{matrix}\right)\] ?????

Answer 5

@Jemurray3

Answer 6

Yes.