Finding Eigenspaces of a Linear Transformation

Example

For $T(x,y,z,w)=(1x,2y,2z,3w)$, find $E_1,E_2,E_3,E_4$ Eigenspace of $T$

For $E_1$

1. We want $E_1=\{v:T(v)=1\ast v\}$
2. $=ker(T-1\ast I)$

$$\left[\begin{array}{ccccc}1-1& 0& 0& 0& 0\\ 0& 2-1& 0& 0& 0\\ 0& 0& 2-1& 0& 0\\ 0& 0& 0& 3-1& 0\end{array}\right]$$

$$\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 2& 0\end{array}\right]$$

5. Solving gives $E_1=\{(x,0,0,0):x\in \mathbb{R}\}$

For $E_2$

1. We want $E_2=\{v:T(v)=2\ast v\}$
2. $=ker(T-2\ast I)$

$$\left[\begin{array}{ccccc}1-2& 0& 0& 0& 0\\ 0& 2-2& 0& 0& 0\\ 0& 0& 2-2& 0& 0\\ 0& 0& 0& 3-2& 0\end{array}\right]$$

$$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right]$$

5. $E_2=\{(0,y,z,0):y,z\in \mathbb{R}\}$

For $E_4$

1. We want $E_4=\{v:T(v)=4\ast v\}$
2. $=ker(T-4\ast I)$

$$\left[\begin{array}{ccccc}1-4& 0& 0& 0& 0\\ 0& 2-4& 0& 0& 0\\ 0& 0& 2-4& 0& 0\\ 0& 0& 0& 3-4& 0\end{array}\right]$$

$$\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\end{array}\right]$$

5. Then $E_4=\{0\}$