0
$\begingroup$

Find a matrix $A$ such that for three vectors $$v_1=(0,-1,1)^T, v_2=(2,1,1)^T,v_3=(-1,1,1)^T$$ $Av_1=1/2v_2$, $Av_2=1/2v_1$ and $Av_3=v_3$.

So we are looking for $A$ such that the matrix representation of $(Av_1,Av_2,Av_3)$ is $$\begin{pmatrix} 0 & 1/2 & 0 \\ 1/2 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}$$ But I m stuck on what to do next. Thanks

$\endgroup$
  • $\begingroup$ Express the canonical vectors $e_1=(1,0,0)$, $e_2=(0,1,0)$ and $e_3=(0,0,1)$ as linear combination of the $v_i$, then compute their images $Ae_i$, and $A=(Ae_1\ Ae_2\ Ae_3)$. $\endgroup$ – Nihl Jul 21 '15 at 9:06
1
$\begingroup$

Note: $[Av_1\ Av_2\ Av_3]=A[v_1\ v_2\ v_3]$. Let $B=[v_1\ v_2\ v_3]$. By assumption, $AB=C$ where $C=\begin{bmatrix} 0& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0\\ 0& 0& 1\end{bmatrix}$. Thus $A=CB^{-1}$. $B^{-1}$ exists, by the way.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.