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: \(L=L^{-1}\).