(Solved): 1. Prove the superposition principle of a linear transformation. Hint: Consider the principle of m ...

1. Prove the superposition principle of a linear transformation. Hint: Consider the principle of mathematical induction. 2. Let \( V \) and \( W \) be vector spaces of the field \( \mathbb{F} \). Do each part as indicated (a) Prove that \( Z_{0}: V \rightarrow W \) is a linear transformation. (b) Prove that \( I_{V}: V \rightarrow V \) is a linear operator on \( V \). 3. Define a function \( T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) according to the rule \[ T(x, y)=(y, x-y) . \] Prove that \( T \) is a linear transformation so that \( T \in \mathscr{L}\left(\mathbb{R}^{2}, \mathbb{R}^{2}\right) \).