References: edX online course MIT 8.05.1x Week 3.
Sheldon Axler (2015), Linear Algebra Done Right, 3rd edition, Springer. Chapter 3.
Having looked at some of the properties of a vector space, we can now look at linear maps. A linear map is defined as a function that maps one vector space into another (possibly the same) vector space , written as
The linear map must satisfy the two properties
- Additivity: for all .
- Homogeneity: for all and all . As usual, the field is either the set of real or complex numbers.
This definition of a linear map is general in the sense that the two vector spaces and can be any two vector spaces. In physics, it’s more common to have , and in such a case, the linear map is called a linear operator.
With a couple of extra definitions, the set of all linear operators on is itself a vector space, with the operators being the vectors. In order for this to be true, we need the following:
- Zero operator: A zero operator, written as just 0 (the same symbol now being used for three distinct objects: the scalar 0, the vector 0 and the operator 0; again the correct meaning is usually easy to deduce from the context) which has the property that the result of acting with 0 on any vector produces the 0 vector. That is , where the 0 on the LHS is the zero operator and the 0 on the RHS is the zero vector.
- Identity operator: An identity operator (sometimes written as 1) leaves any vector unchanged, so that for all .
With these definitions, is now a vector space, since it satisfies the distributive (additivity) and scalar multiplication (homogeneity) properties, contains an additive identity (the zero operator) and a multiplicative identity (the identity operator).
In addition, there is a natural definition of the multiplication of two linear operators and , written as . When a product operates on a vector , we just operate from right to left in succession, so that
The product of two operators produces another operator also in , since this product also satisfies additivity and homogeneity:
A very important property of operator multiplication is that it is not commutative. We’ve already seen many examples of this in our journey through quantum mechanics with operators such as position and momentum, angular momentum and so on. The non-commutativity is a fundamental mathematical property however, and can be seen in other examples that have nothing to do with quantum theory.
For example, consider the left shift operator and right shift operator , defined to act on the vector space consisting of infinite sequences of numbers. That is, our vector space is such that
where . The shift operators have the following effects:
The operator removes the first element in the sequence, while the operator inserts a 0 (number!) as the new first element in the sequence. Note that 0 is the only number we could insert into a sequence in order that be a linear operator, since from additivity above, we must have . That is, if we start with (the vector all of whose elements ), then must also give the zero vector.
The two products and produce different results:
The difference is called the commutator of the two operators and . If we introduce the operator which projects out the first element in the sequence:
then we have