Answer the following questions: 1. Let 𝑉 = ℝ3 . Determine the following subsets are subspaces or not. a) 𝑊 = 𝑥, 𝑦, 𝑧 : 2𝑥 − 7𝑦 + 𝑧 = 0 b) 𝑊 = 𝑥, 𝑦, 𝑧 : 2𝑥 − 3𝑦 + 𝑧 = 3 c) 𝑊 = 𝑥, 𝑦, 𝑧 : 𝑥 = 3𝑦 𝑎𝑛𝑑 𝑧 = −𝑦 𝑑2𝑦 𝑑𝑦 2. Let 𝑊 be the set of all solutions of the differential equation 𝑑 𝑥 2 − 3 𝑑𝑥 + 2𝑦 = 0. Prove that 𝑊 is a vector space. 3. Find a basis and dimension of the subspace a) 𝑆 = 𝑥, 𝑦, 𝑧 𝜖ℝ3 : 2𝑥 + 𝑦 − 𝑧 = 0 . b) 𝑆 = 𝑥, 𝑦, 𝑧, 𝑤 𝜖ℝ4 : 𝑥 + 𝑦 + 𝑧 + 𝑤 = 0 . 4. Prove that the following set S is a subspace of set of all real matrices of size 2X2. Find basis of S and dim(S). 𝑆= 𝑎 𝑏 5. If 𝑆 = 𝑥, 𝑦, 𝑧 𝜖ℝ3 : 𝑥 + 𝑦 + 𝑧 = 0 , 𝑇 = dim 𝑆 , dim 𝑇 , dim (𝑆 ∩ 𝑇). 𝑏 : 𝑎, 𝑏, 𝑐 ∈ ℝ . 𝑐 𝑥, 𝑦, 𝑧 𝜖ℝ3 : 𝑥 + 2𝑦 − 𝑧 = 0 , find 𝑎 𝑏 𝑎 𝑏 : 𝑎, 𝑏, 𝑐 ∈ ℝ, 𝑎 + 𝑏 = 0 , 𝑇 = : 𝑎, 𝑏, 𝑐 ∈ ℝ, 𝑐 + 𝑑 = 0 are two 𝑐 𝑑 𝑐 𝑑 subspaces of 𝑀2 (ℝ), find dim 𝑆 , dim 𝑇 , dim (𝑆 ∩ 𝑇). 6. If 𝑆 = 7. Find the dimension of the subspace spanned by 1,0, −1 , 1,8,14 , (0, −4,3) in ℝ3 . 8. Find the dimension of the subspace spanned by 1,1,2,4 , 2, −1, −5,2 , (1, −1, −4,0) and (2,1,1,6) in ℝ4 .