(Solved): please 4. (8 points) Let \( V \) and \( W \) be vector spaces over \( \mathbb{R} \) and \( T: V \r ...

please

4. (8 points) Let \( V \) and \( W \) be vector spaces over \( \mathbb{R} \) and \( T: V \rightarrow W \) a linear transformation. Let \( \left\{v_{1}, \ldots, v_{n}\right\} \) be a basis for \( V \). (a) Prove that \( \left\{T\left(v_{1}\right), \ldots, T\left(v_{n}\right)\right\} \) is a spanning set for range \( (T) \). (In your argument, indicate clearly where you are using the facts that (i) \( \left\{v_{1}, \ldots, v_{n}\right\} \) is a basis for \( V \) and (ii) \( T \) is linear.) (b) Give a concrete example of vector spaces \( V \) and \( W \), a basis \( \left\{v_{1}, \ldots, v_{n}\right\} \) of \( V \), and linear transformation \( T \) such that \( \left\{T\left(v_{1}\right), \ldots, T\left(v_{n}\right)\right\} \) is not a basis for range \( (T) \).