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 .