Reference no: EM133327982
Question 1. Consider the vector space V = C3 with the scalar field C and addition and scalar multiplication defined as follows:
(x1, y1, z1) + (x2, y2, z2) = (x1 + x2 + 7, y1 + y2, z1 + z2 - 4) for any (x1, y1, z1), (x2, y2, z2) ∈ V and
α(x1, y1, z1) = (αx1 + 7α - 7, αy1, αz1 - 4α + 4)
for any α ∈ C. This defines a vector space. (You do not have to prove this!)
(a) Find the zero vector for V.
(b) Find the additive inverse of (x , y , z ) in V.
Question 2. (NS 1.2.12b) Let U and W be subspaces of a vector space V. Prove that U ∪ W is a subspace of V if and only if U ⊆ W or W ⊆ U.
Question 3. Let V = Cn×n for n ≥ 2 be the complex vector space with the usual matrix addition and scalar multiplication. Let U1 = {A ∈ Cn×n: AT = A} be the set of symmetric matrices in V and let U2 = {A ∈ Cn×n : A* = A} be the set of hermitian matrices in V . One of U1 and U2 is a subspace of V while the other is not. Determine which is which.
Question 4. Find a basis for the real vector space
V = {(a, b, c, d ) ∈ R4: c = -6b, d = 2a}
with the usual addition and scalar multiplication.
Question 5. Under what conditions on α is
{(1, α, 0), (α, 0, 1), (1 - α, α, 1)}
a basis for R3?