How to complete basis? Two vectors
$a = (1,-2,2,-3)$ and $b=(2,-3,2,4)$
I need to complete the basis for $\Bbb{R}^4$
I know that there's a way of solving 2-equations linear system like $$\begin{pmatrix}a \\ b\ \end{pmatrix}\cdot x = 0$$ to get vector $c$, and then solving 3-equations linear system to get vector $d$
But are not there other ways? I heard that gram schmid proccess could be helpful, but I am not sure how to apply it here as well ass if this idea is correct at all.
 A: One way to find other two vectors is to consider some simple vectors, such as $(1,0,0,0)$, $(0,1,0,0)$ etc. and check whether they are linearly dependent on $a$ and $b$ or not.
A: Your idea is correct: that corresponds to put $\mathbf a \cdot \mathbf c= \mathbf b \cdot \mathbf c=0$. Only that , when you do it again for $\mathbf d$, either you impose also $\mathbf c \cdot \mathbf d = 0$, or you check that  the found $\mathbf c$ and $\mathbf d$ be independent.   
Actually, imposing  $\mathbf c$ to be normal to $\mathbf a$ and $\mathbf b$ is too a strict condition: it would be sufficient that they are mutually independent. For that you can construct a $4 \times 4$ matrix, with the first two columns occupied by $\mathbf a$ and $\mathbf b$, and then try and fill the other columns as suggested in the previous answer, checking that the determinant be non-null.
The Gram-Schmidt process is not applicable in your problem. It is useful, once you have $4$ independent vector, to construct from them a normal basis.
