- #1
squenshl
- 479
- 4
- Homework Statement
- Let ##\mathbb{R}^3\mapsto \mathbb{R}^3## be given by $$T\begin{pmatrix}
x \\
y \\
z
\end{pmatrix} := \begin{pmatrix}
x+y-2z \\
2x-2z \\
y-x
\end{pmatrix}.$$
1. Give a basis of eigenvectors for ##\text{ker}(T)##.
2. Give a basis of eigenvectors for ##\text{ran}(T)##.
- Relevant Equations
- None
Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v =
\begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
1 \\
0
\end{pmatrix}
##.
Not sure what to do next. Thanks!
\begin{pmatrix}
1 \\
1 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
0 \\
1
\end{pmatrix},
\begin{pmatrix}
1 \\
1 \\
0
\end{pmatrix}
##.
Not sure what to do next. Thanks!