Reference no: EM132550129

Question 1. Show that the positive quadrant Q = {(x, y) | x, y > 0} ⊂ R² forms a vector space if we define addition by (x₁, y₁) + (x₂, y₂) = (x₁x₂, y₁y₂), and scalar multiplication by t (x, y) = (x^t, y^t), where x_i, y_i ∈ R for i ∈ {1, 2} and t ∈ R.

Question 2. Let V be the vector space of all real-valued functions defined on R. Let W be the set of all functions f ∈ V satisfying the following conditions: there exist a, b ∈ R+ such that for all x ∈ R satisfying |x| ≥ a we have |f (x) |≤ b |x| . Show that W is a subspace of V .

Question 3. Determine whether the following subsets are subspaces of the given vector space V .
(a) W₂ = {(a + 2b, a, 2a - b) | a, b ∈ R} where V = R³
(b) W₃ = {(x₁, x₂, x₃) ∈ R³ | x₁x₂x₃ = 0} where V = R³
(c) W₄ = {a₂t² + a₁t + a₀ | a₂ + a₁ = a₀} where V = P₂(R)
(d) W₅ = {(a b c)| a + c = 2e + 1} where V = M_2x3(R)
             {(d e f)|

Question 4. Let V be a vector space over a field F. Let V₁ and V₂ be subspaces of V .
(a) Define V₁ + V₂ as V₁ + V₂ = {v₁ + v₂ | v₁ ∈ V₁, v₂ ∈ V₂}. Show that V₁ + V₂ is a subspace of V .
(b) Show that V₁ ∩ V₂ is a subspace of V
(c) Prove that V₁ U V₂ is a subspace of V if and only if one of the subspaces is contained in the other.

Question 5. Determine which of the following subsets A, B of R³ are linearly dependent or inde- pendent. Which of the subsets span R³?
(a) A = {(1, 1, 1), (1, 2, 3), (2, -1, 1)} (b) B = {(1, 1, 2), (1, 2, 5), (5, 3, 4)}

Question 6. Let V be a vector space over a field F. Let B₁ span V and suppose B₂ is a set such that if h ∈ B₁, then h = α₁b₁ + .... + α_kb_k, for some b₁, . . . , b_k ∈ B₂ and α₁, . . . , α_k ∈ F . Prove that B₂ also spans V .

Question 7. Determine whether the given function is a linear transformation.
(a) L : P₂ (R) -→ P₃ (R)is defined by L(f (x)) = xf (x) + f'(x)
(b) g : R³ -→ R³ is defined by g (x, y, z) = (2x, y - 2, 3y).
(c) φ : M (R) -→ M₂₂ (R) is defined by φ (p q r) Σ = (2p - q 2q + r)

                                                         (u v w)     (0     0)

Question 8. Let V and W be vector spaces over a field F . Prove that if a linear transformation f : V -→ W is one-to-one then, {f (v₁), f (v₂) , . . . , f (v_n)} is a linearly independent subset of W whenever {v_1, v₂, . . . , v_n} is a linearly independent subset of V .

Question 9. Find F (a, b), where the linear map F : R² → R² is defined by F (1, 2) = (3, -1) and F (0, 1) = (2, 1).

Question 10. Let k : P₁(R) R² be a linear map such that k(x+1) = (1, 1) and k(x 1) = (0, 1). Find k(x + 3).