- #1
chrisb93
- 8
- 0
Homework Statement
It's given or I've already shown in previous parts of the question:
[itex]A \in M_{nxn}(F)\\
A^{2}=I_{n}\\
F = \mathbb{Q}, \mathbb{R} or \mathbb{C}\\
ker(L_{I_{n}+A})=E_{-1}(A)[/itex]
Eigenvalues of A must be [itex]\pm1[/itex]
Show [itex]im(L_{I_{n}+A})=E_{1}(A)[/itex] where E is the eigenspace for the eigenvalue 1
(I also need to show that [itex]im(L_{I_{n}-A})=E_{-1}(A)[/itex] but I think that should be simple once I've done one of them)
Homework Equations
The Attempt at a Solution
I know that I need to show both sets are contained within the other set so,
Show [itex]im(L_{I_{n}+A}) \subseteq E_{1}(A)[/itex]
[itex]y=L_{I_{n}+A}(x)[/itex] Let y be a general element of the image
[itex]=x+Ax[/itex] By definition of the transformation
[itex]\Rightarrow A y = A x + A^{2} x[/itex] Multiply through by A
[itex]= A x + x [/itex] As A2 is the identity element
[itex]\Rightarrow A y = y \in E_{1}(A)[/itex] As [itex]E_{1}(A) := \{ x | A x = x \}[/itex]
I've no idea how to show [itex]E_{1}(A) \subseteq im(L_{I_{n}+A})[/itex]