Proving Linear Operators: Commutator Relationships

[blanik]

I'm not sure where to start with these proofs. Any suggestions getting started would be appreciated.

1. Show that is A,B are linear operators on a complex vector space V, then their product (or composite) C := AB is also a linear operator on V.

2. Prove the following commutator relationships for Linear Operators A,B,C:
a. [A,B + C] = [A,B] + [A,C]
b. [A,B] = -[B,A]
c. [A,BC] = B[A,C] + [A,B]C
d. [AB,C] = A[B,C] + [A,C]B

 

[Hurkyl]

I'm not sure where to start

Definitions.

 

[thepatientmental]

To start with these proofs, it would be helpful to have a clear understanding of what a linear operator is and how it operates on a vector space. A linear operator is a function that maps vectors from one vector space to another, while preserving the vector space structure. In other words, it satisfies the properties of additivity and homogeneity.

For the first proof, we need to show that the product of two linear operators, A and B, is also a linear operator. To do this, we need to show that C := AB satisfies the properties of additivity and homogeneity. Let x and y be vectors in V and c be a complex number. Then we have:

C(x + y) = A(B(x + y)) = A(Bx + By) (since B is a linear operator)
= A(Bx) + A(By) (by the property of additivity of A)
= Cx + Cy (since A is also a linear operator)

Similarly, we have:

C(cx) = A(B(cx)) = A(cBx) (since B is a linear operator)
= cA(Bx) (by the property of homogeneity of A)
= cCx (since A is also a linear operator)

Therefore, C satisfies the properties of additivity and homogeneity, and hence is a linear operator on V.

For the second proof, we need to show the following commutator relationships:

a. [A,B + C] = [A,B] + [A,C]
b. [A,B] = -[B,A]
c. [A,BC] = B[A,C] + [A,B]C
d. [AB,C] = A[B,C] + [A,C]B

To prove these, we will use the definition of commutator, which is given by [A,B] := AB - BA. Let x be a vector in V. Then we have:

a. [A,B + C]x = (B + C)x - x(B + C) (by definition)
= Bx + Cx - (Bx + Cx) (since B and C are linear operators)
= Bx + Cx - Bx - Cx (by the property of additivity)
= Bx - Bx + Cx - Cx (by the property of additivity)
= [A,B]x + [A,C]x (by definition)