Let V be a finite-dimensional vector space and let T: V → V be a linear map such that T² = T. 1. Show that V = Null(T) ⊕ Range(T). In other words, prove that every vector in V can be written uniquely as a sum of a vector in Null(T) and a vector in Range(T). 2. Show that T is the projection onto Range(T) along Null(T). Here's a brief outline of the proof to give you a starting point: 1. To prove V = Null(T) ⊕ Range(T), you need to show two things: firstly, that every vector v in V can be written as a sum of a vector in Null(T) and a vector in Range(T); and secondly, that this representation is unique. The first part is typically proven using the fact that T² = T, while the second part usually involves some argument based on the properties of the null space and range. 2. To prove that T is the projection onto Range(T) along Null(T), you need to show that for any vector v in V, T(v) is the unique vector in Range(T) such that v - T(v) is in Null(T). This can be proven using the definition of a projection and the properties of T, Null(T), and Range(T). Remember that the details of the proofs can vary depending on the exact properties of V and T that you're allowed to use.