Linear Maps

Null Spaces and Ranges Exercise 1 Give an example of a linear map \( T \) such that \(\dim \text{null} \, T = 3\) and \(\dim \text{range} \, T = 2\). Exercise 2 Suppose \( V \) is a vector space and \( S, T \in \mathcal{L}(V, V) \) are such that \[\text{range} \, S \subset \text{null} \, T.\] Prove that \((ST)^2 = 0\). Exercise 3 Suppose \( v_1, \ldots, v_m \) is a list of vectors in \( V \). Define \( T \in \mathcal{L}(\mathbb{F}^m, V) \) by ...

Vector Space of Linear Maps Exercise 1 Suppose \( b, c \in \mathbb{R} \). Define \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) by \[T(x, y, z) = (2x - 4y + 3z + b, 6x + cxyz).\] Show that \( T \) is linear if and only if \( b = c = 0 \). Exercise 2 Suppose \( b, c \in \mathbb{R} \). Define \( T: \mathcal{P}(\mathbb{R}) \to \mathbb{R}^2 \) by ...