<s>[INST] find a basis for $R^4$ given the vectors. i need help with this excercise..
find a basis for $R^4$ given the vectors.
$$\begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \\ 0  \end{pmatrix}\quad \text{and}\quad \begin{pmatrix} -\frac{1}{2} \\ \frac{1}{2} \\ \frac{1}{2} \\ -\frac{1}{2}  \end{pmatrix}$$ 
What method do I follow to find the other two vectors?
 [/INST] Clearly, $v_1$ and $v_2$ are linearly independent, then you can add a new vector $v_3$ that is not in the span of 
$$v_1=\begin{pmatrix} \frac{1}{\sqrt{2}} \\ 0 \\ \frac{1}{\sqrt{2}} \\ 0  \end{pmatrix}\quad \text{and}\quad v_2=\begin{pmatrix} -\frac{1}{2} \\ \frac{1}{2} \\ \frac{1}{2} \\ -\frac{1}{2}  \end{pmatrix}$$
Finally add a 4th vector $v_4$ that is not an element in the span of $v_1,v_2,v_3$.
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