To prove that \( T \) is a linear map if and only if the graph of \( T \) is a subspace of \( V \times W \), we need to show two things: 1. If \( T \) is a linear map, then its graph is a subspace of \( V \times W \). 2. If the graph of \( T \) is a subspace of \( V \times W \), then \( T \) is a linear map. Let's start with the first part: 1. If \( T \) is a linear map, then its graph is a subspace of \( V \times W \). To prove this, let's assume \( T \) is a linear map. We want to show that the graph of \( T \) is closed under addition and scalar multiplication, which are the defining properties of a subspace. Let \( (v_1, Tv_1) \) and \( (v_2, Tv_2) \) be two arbitrary elements of the graph of \( T \). Since \( T \) is linear, we have: \[ T(v_1 + v_2) = Tv_1 + Tv_2 \] So, \( (v_1 + v_2, Tv_1 + Tv_2) \) is also in the graph of \( T \), meaning the graph is closed under addition. Similarly, for any scalar \( \alpha \), we have: \[ T(\alpha v_1) = \alpha Tv_1 \] So, \( (\alpha v_1, \alpha Tv_1) \) is also in the graph of \( T \), showing closure under scalar multiplication. Since the graph of \( T \) is closed under addition and scalar multiplication, it is indeed a subspace of \( V \times W \). Now, let's prove the second part: 2. If the graph of \( T \) is a subspace of \( V \times W \), then \( T \) is a linear map. To prove this, let's assume the graph of \( T \) is a subspace of \( V \times W \). We need to show that \( T \) satisfies the properties of linearity. Let \( v_1, v_2 \) be arbitrary vectors in \( V \), and \( \alpha \) be an arbitrary scalar. We want to show that \( T(v_1 + v_2) = T(v_1) + T(v_2) \) and \( T(\alpha v_1) = \alpha T(v_1) \). Since \( (v_1, Tv_1) \) and \( (v_2, Tv_2) \) are in the graph of \( T \), their sum \( (v_1 + v_2, Tv_1 + Tv_2) \) must also be in the graph, by closure under addition. Therefore, \[ T(v_1 + v_2) = Tv_1 + Tv_2 \] Similarly, for scalar multiplication, since \( (\alpha v_1, \alpha Tv_1) \) is in the graph of \( T \) (by closure under scalar multiplication), we have: \[ T(\alpha v_1) = \alpha Tv_1 \] Thus, \( T \) satisfies the properties of linearity. Therefore, we have shown both directions of the proof, concluding that \( T \) is a linear map if and only if the graph of \( T \) is a subspace of \( V \times W \).