For (a1, a2), (b1, b2) ∈ V and c ∈ R,define (a1, a2) + (b1, b2) = (a1 + 2b1, a2 + 3b2) and c(a1, a2) = (ca1, ca2). Is V a vector space over R with these operations? Examine all the properties and …

Solution for Problem 2. Let V = {(a₁, a₂): a1, a2 ER}. For (a₁, a2), (b₁,b₂) € V and c E R, define (a1, a2) + (b₁,b2) = (a₁ + 2b₁, a2 + 3b₂) and c ...

Define a sequence a1, a2, a3, . . . as follows: a1 = 1, a2 = 3, and a = ak-1+ak-2 for every %D %3D integer k > 3. (This sequence is known as the Lucas sequence.) Use strong …

Solution for 4 Problem 4. Let V = {(a1, a2) a1, a2 in R}; that is, V is the set consisting of all ordered pairs (a1, a2), where a₁ and a2 are real numbers.

a) Prove that A1*= A1, A2*=A2, and A=A1+iA2. Would it be reasonable to define A1 and A2 to be the real and imaginary parts, respectively, of the matrix A ? b) Let A be an n x n matrix.

Solution for (1) (a) Define A1 := {A CR : A is open} and A2 := {A CR: A is closed}. Is Aj is a o-algebra? Is Az is a o-algebra?

a) Let V = {(a₁, a2): a1, a2 € R}. For (a₁, a2), (b₁,b₂) EV and c € R, define: (a1, a2)+(b₁,b₂) = (a₁ +2b1, a2 +3b2) and c(a1, a2) = (ca₁,ca2) Determine whether the following sets with the …