First, we define $B=\{u_1,u_2,u_3\}$ as a basis of $\mathbb{R^3}$.
We also have the linear transformation $T \in L(\mathbb{R^3})$, whose kernel is defined by $$ker(T)=\{(x_1,x_2,x_3)/x_1+x_2-x_3=0,x_2+x_3=0\}$$ Two associated eigenvalues are $1$ and $-1$, with the respective eigenvectors $u_1+u_2-u_3$ and $u_2+u_3$. Find the matrix of $T$ associated to the basis $B$.
The only thing I managed to find that might be useful is a basis for the kernel. $$ker(T)=\left\{\begin{bmatrix}2 \\-1\\1\end{bmatrix} \right\}$$ How should I proceed from there?

  • $\begingroup$ Try eigen decomposition $\endgroup$ – Henricus V. Oct 18 '15 at 23:15
  • $\begingroup$ Wouldn't I need to know all three eigenvalues and their respective eigenvectors for that? $\endgroup$ – SharkFin Oct 18 '15 at 23:24
  • $\begingroup$ One of the eigenvalues may have a multiplicity higher than $1$. $\endgroup$ – Henricus V. Oct 18 '15 at 23:25
  • $\begingroup$ But how can I be sure if one does, and how would I find another, linearly independent vector associated to that eigenvalue? $\endgroup$ – SharkFin Oct 18 '15 at 23:31
  • $\begingroup$ Test the resulting matrix against your conditions. That is, see if it has the kernel you specify. $\endgroup$ – Henricus V. Oct 18 '15 at 23:46

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