How can I extend a set to an orthogonal basis? Consider the set $\{(1, −2, 3, 4),(1, 1, −1, 1)\}$.
If I wanted to extend this to an orthogonal basis of $\mathbb{R}^4$, I am somewhat confused on the steps to take.
I understand that I would begin by taking the reduced row echelon form of the set's matrix, but after that I am unsure how to extend it, let alone to an orthogonal basis. 
I appreciate any help!
 A: You may apply the Gram-Schmidt orthogonalization process.
The method is based on the choice of some arbitrary basis $\mathcal{B} = \{v_{1},v_{2},\ldots,v_{n}\}$ of the underlying vector space. Then you build an orthogonal basis through the following process: you set $w_{1} = v_{1}$ and obtain $w_{k}$ recursively according to the relation
\begin{align*}
w_{k} = v_{k} - \sum_{j=1}^{k-1}\frac{\langle v_{k},w_{j}\rangle}{\|w_{j}\|^{2}}w_{j}
\end{align*}
The idea is to take $w_{2}$ as the vector $v_{2}$ minus its projection on direction $w_{1}$, the vector $w_{3}$ as the vector $v_{3}$ minus its projection on the directions $w_{1}$ and $w_{2}$ and so on. Since each $w_{k}$ is linear combination of the vectors $v_{1},v_{2},\ldots,v_{n}$, the proposed algorithm does not change the vector space spanned by $\mathcal{B}$.
At your case, you already have two LI vectors. It suffices, then, to add two more vectors in order to obtain a basis and to apply such procedure.
With the purpose of doing so, choose arbitrarily a vector in $\textbf{R}^{4}$ and check if it is a linear combination of $(1,-2,3,4)$ and $(1,1,-1,1)$. If it is, you discard it and choose another one. Do it until you obtain four LI vectors.
Hopefully this helps.
