A linear functional is a special type of linear map that takes a vector as input and produces a scalar (a single number) as output.
We will study linear functionals as a precursor to the more general concept of multilinear maps.
A linear functional is a linear map where the second vector space is the field of scalars itself. The set of all linear functionals from a vector space to its field of scalars is called the dual space of , denoted as ;
Since they are so common, there are many names for linear functionals. They include covectors, dual vectors, bra vectors (with Dirac notation), and one-forms (in differential geometry).
Let be a vector in an -dimensional complex vector space.
The sum of all components of is a linear functional defined as
Similarly, let be a complex matrix. The sum of all entries of is a linear functional defined as
If we just take the diagonal entries of and sum them up, we get another linear functional called the trace, defined as
Some functionals are linear functionals over function spaces. Specifically, let . Then, the linear functional defined as
is the definite integral of over the interval .
There is a duality between inner products and linear functionals. In particular, given an inner product space with inner product , for each fixed vector , we can define a linear functional as
We shall show that this duality exists between vectors and linear functionals, between vector spaces and dual spaces, and between linear maps and their dual maps.
There exists a linear isomorphism between a finite-dimensional vector space and its dual space . We can think of a linear functional as a row vector that acts on a column vector from the left via matrix multiplication to produce a scalar:
More accurately, a vector is represented itself as a product of a column vector and the standard basis row vectors:
where is the standard basis of .
Similarly, we can define a basis for the dual space consisting of linear functionals such that
where is the Kronecker delta.
Written in matrix form, we have
Let be a finite-dimensional vector space over the field with basis , with basis .
Then, its dual space is also finite-dimensional with the same dimension as , and there exists a basis of such that
From this theorem, it is clear that for every vector in , there exists a unique corresponding linear functional in , and vice versa. Specifically, for each vector , we can define a corresponding linear functional as
Let be a vector space over the field , and let be a vector.
The annihilator of is a linear functional such that .
The space of all annihilators of a subspace is called the annihilator of , denoted as .
is a subspace of the dual space . To see this, let be two annihilators of , and let be two scalars. Then, for any vector , we have
Thus, is also an annihilator of , and hence is closed under addition and scalar multiplication.
Also, suppose has basis . We can always extend its basis to a basis of by adding vectors . We can show that the dual basis corresponding to the added vectors forms a basis for .
To see this, let be an annihilator of . Then, we can express as
for some scalars .
Since annihilates all vectors in , we have
for all .
Thus, , and hence
This shows that spans . Moreover, these functionals are linearly independent by definition of the dual basis.
Therefore, forms a basis for . It also follows that
Let be a linear map between two vector spaces and over the field . The dual map (or pullback) of is a linear map defined as
for all and .
Perhaps it helps to provide a concrete example. Suppose we have and a linear map defined via the matrix . Then, we have .
Now, let be a linear functional defined via the row vector . So . The pullback is then a linear functional defined as , so its matrix representation is given by
One can verify that , which is consistent with the definition of the dual map.
There are a few important properties of the dual map worth mentioning.
The dual map is linear. This is easily verified from the definition.
if and only if annihilates the image of . To see this, let . Then, we have
By the definition of the dual map, this is equivalent to
but this just means that for any vector in the image of , maps it to zero, i.e., annihilates the image of .
Notationally, (the annihilator of the image of ).
If is surjective (i.e., ), then annihilates all of . But this just means that is the zero functional. Thus, we have shown that if is surjective, then , which means that is injective (Theorem 2.3.11).
We shall summarize these properties in a proposition.
Let be a linear map between two vector spaces and over the field , and let be its dual map. Then, . In particular, if is surjective, then is injective. If is an isomorphism, then so is .
In accordance with Dirac notation, we make some notational changes to make the duality between vectors and linear functionals more explicit. Let be a basis for , and let the scalars uniquely define a linear functional such that .
Since , we identify with the symbol , written as the association .
It's common to refer to as a bra vector, in contrast to the ket vector . We often use the notation
where the is called a dagger or dual operation. Let's see how applies to linear combinations. Let be a linear combination of two vectors . We can take the inner product of with any basis vector to get
Taking the complex conjugate of both sides, we have
so
In other words, we have the first association , which is linear, and the second association , which is sesquilinear. In matrix form, we have