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Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2$ matrix.

a) If $L$ is invertible, then $L^{-1}$ is also invertible.
b) If $L V=5 V$ for all vectors $V$, then $L^{-1} W=(1 / 5) W$ for all vectors $W$.
c) If $L$ is a rotation of the plane by 45 degrees counterclockwise, then $L^{-1}$ is a rotation by 45 degrees clockwise.
d) If $L$ is a rotation of the plane by 45 degrees counterclockwise, then $L^{-1}$ is a rotation by 315 degrees counterclockwise.
e) The zero map $(0 \mathbf{V}=0$ for all vectors $\mathbf{V})$ is invertible.
f) The identity map ( $I \mathbf{V}=\mathbf{V}$ for all vectors $\mathbf{V}$ ) is invertible.
g) If $L$ is invertible, then $L^{-1} 0=0$.
h) If $L \mathbf{V}=0$ for some non-zero vector $\mathbf{V}$, then $L$ is not invertible.
i) The identity map (say from the plane to the plane) is the only linear map that is its own inverse: $L=L^{-1}$.
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