<s>[INST] Vectors as linear combinations I want to express v=$(\sqrt2,1/2)$ as linear combination of p=(1,2) and q=(2,1).
My answer:
$(\sqrt2,1/2)$=c(1,2)+d(2,1). So solve for c and d:
$\begin{pmatrix} \sqrt2 \\ 1/2 \end {pmatrix}=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}\begin{pmatrix} c \\ d\end{pmatrix}$
$\begin{pmatrix} c \\ d \end {pmatrix}=\begin{pmatrix} -\sqrt2/3 +1/3 \\ 2\sqrt2/3+1/6 \end{pmatrix}$
Is this right? It doesn't feel right for some reason. 
 [/INST] We have
$$
\binom{c}{d}=\begin{pmatrix} 1 & 2 \\ 2 & 1\end{pmatrix}^{-1}\binom{\sqrt{2}}{1/2}=\frac{1}{3}\begin{pmatrix} -1 & 2 \\ 2 & -1\end{pmatrix}\binom{\sqrt{2}}{1/2}=\frac{1}{3}\binom{1-\sqrt{2}}{(1-4\sqrt{2})/2}
$$
</s>