In this section, we are concerned exclusively with linear transformations from a vector space V into its field of scalars F, which is itself a vector space of dimension 1 over F. Such a linear transformation is called a linear functional on V. We generally use the letters f, g, h, ... to denote linear functionals. As we see in Example 1, the definite integral provides us with one of the most important examples of a linear functional in mathematics.
Let V be the vector space of continuous real-valued functions on the interval . Fix a function . The function defined by
is a linear functional on V. In the cases that g(t) equals sin nt or cos nt, h(x) is often called the nth Fourier coefficient of x.
Let , and define by , the trace of A. By Exercise 6 of Section 1.3, we have that f is a linear functional.
Let V be a finite-dimensional vector space, and let be an ordered basis for V. For each , define , where
is the coordinate vector of x relative to . Then is a linear functional on V called the ith coordinate function with respect to the basis . Note that , where is the Kronecker delta. These linear functionals play an important role in the theory of dual spaces (see Theorem 2.24).
For a vector space V over F, we define the dual space of V to be the vector space L(V, F), denoted by V*.
Thus V* is the vector space consisting of all linear functionals on V with the operations of addition and scalar multiplication as defined in Section 2.2. Note that if V is finite-dimensional, then by the corollary to Theorem 2.20 (p. 105)
Hence by Theorem 2.19 (p. 104), V and V* are isomorphic. We also define the double dual V** of V to be the dual of V*. In Theorem 2.26, we show, in fact, that there is a natural identification of V and V** in the case that V is finite-dimensional.
Suppose that V is a finite-dimensional vector space with the ordered basis . Let be the ith coordinate function with respect to as just defined, and let . Then is an ordered basis for V*, and, for any , we have
Let . Since , we need only show that
from which it follows that generates V*, and hence is a basis by Corollary 2(a) to the replacement theorem (p. 48). Let
For , we have
Therefore by the corollary to Theorem 2.6 (p. 73).
Using the notation of Theorem 2.24, we call the ordered basis of V* that satisfies the dual basis of .
Let be an ordered basis for . Suppose that the dual basis of is given by . To explicitly determine a formula for , we need to consider the equations
Solving these equations, we obtain and ; that is, . Similarly, it can be shown that .
Now assume that V and W are n- and m-dimensional vector spaces over F with ordered bases and , respectively. In Section 2.4, we proved that there is a one-to-one correspondence between linear transformations and matrices (over F) via the correspondence . For a matrix of the form , the question arises as to whether or not there exists a linear transformation U associated with T in some natural way such that U may be represented in some basis as . Of course, if , it would be impossible for U to be a linear transformation from V into W. We now answer this question by applying what we have already learned about dual spaces.
Let V and W be finite-dimensional vector spaces over F with ordered bases and , respectively. For any linear transformation , the mapping defined by for all is a linear transformation with the property that .
For , it is clear that is a linear functional on V and hence is in V*. Thus maps W* into V*. We leave the proof that is linear to the reader.
To complete the proof, let and with dual bases and , respectively. For convenience, let . To find the jth column of , we begin by expressing as a linear combination of the vectors of . By Theorem 2.24, we have
So the row i, column j entry of is
Hence .
The linear transformation defined in Theorem 2.25 is called the transpose of T. It is clear that is the unique linear transformation U such that .
We illustrate Theorem 2.25 with the next example.
Define by . Let and be the standard ordered bases for and , respectively. Clearly,
We compute directly from the definition. Let and . Suppose that . Then . So
But also
So . Using similar computations, we obtain that , and . Hence a direct computation yields
as predicted by Theorem 2.25.
We now concern ourselves with demonstrating that any finite-dimensional vector space V can be identified in a natural way with its double dual V**. There is, in fact, an isomorphism between V and V** that does not depend on any choice of bases for the two vector spaces.
For a vector , we define by for every . It is easy to verify that is a linear functional on V*, so . The correspondence allows us to define the desired isomorphism between V and V**.
Let V be a finite-dimensional vector space, and let . If for all , then .
Let . We show that there exists such that . Choose an ordered basis for V such that . Let be the dual basis of . Then . Let .
Let V be a finite-dimensional vector space, and define by . Then is an isomorphism.
(a) is linear: Let . For , we have
Therefore
(b) is one-to-one: Suppose that is the zero functional on V* for some . Then for every . By the previous lemma, we conclude that .
(c) is an isomorphism: This follows from (b) and the fact that .
Let V be a finite-dimensional vector space with dual space V*. Then every ordered basis for V* is the dual basis for some basis for V.
Let be an ordered basis for V*. We may combine Theorems 2.24 and 2.26 to conclude that for this basis for V* there exists a dual basis in V**, that is, for all i and j. Thus is the dual basis of .
Although many of the ideas of this section (e.g., the existence of a dual space) can be extended to the case where V is not finite-dimensional, only a finite-dimensional vector space is isomorphic to its double dual via the map . In fact, for infinite-dimensional vector spaces, no two of V, V*, and V** are isomorphic.
Label the following statements as true or false. Assume that all vector spaces are finite-dimensional.
(a) Every linear transformation is a linear functional.
(b) A linear functional defined on a field may be represented as a matrix.
(c) Every vector space is isomorphic to its dual space.
(d) Every vector space is isomorphic to the dual of some vector space.
(e) If T is an isomorphism from V onto V* and is a finite ordered basis for V, then .
(f) If T is a linear transformation from V to W, then the domain of is V**.
(g) If V is isomorphic to W, then V* is isomorphic to W*.
(h) The derivative of a function may be considered as a linear functional on the vector space of differentiable functions.
For the following functions f on a vector space V, determine which are linear functionals.
(a) where′ denotes differentiation
(b)
(c)
(d)
(e)
(f)
For each of the following vector spaces V and bases , find explicit formulas for vectors of the dual basis for V*, as in Example 4.
(a)
(b)
Let , and define as follows:
Prove that is a basis for V*, and then find a basis for V for which it is the dual basis.
Let , and, for , define by
Prove that is a basis for V*, and find a basis for V for which it is the dual basis.
Define by and by .
(a) Compute .
(b) Compute , where is the standard ordered basis for and is the dual basis, by finding scalars a, b, c, and d such that and .
(c) Compute and , and compare your results with (b).
Let and with respective standard ordered bases and . Define by
where is the derivative of p(x).
(a) For defined by , compute .
(b) Compute without appealing to Theorem 2.25.
(c) Compute and its transpose, and compare your results with (b).
Let be a linearly independent set in . Show that the plane through the origin in may be identified with the null space of a vector in .
Prove that a function is linear if and only if there exist such that for all . Hint: If T is linear, define for ; that is, for , where is the dual basis of the standard ordered basis for .
Let , and let be distinct scalars in F.
(a) For , define by . Prove that is a basis for V*. Hint: Apply any linear combination of this set that equals the zero transformation to , and deduce that the first coefficient is zero.
(b) Use the corollary to Theorem 2.26 and (a) to show that there exist unique polynomials such that for . These polynomials are the Lagrange polynomials defined in Section 1.6.
(c) For any scalars (not necessarily distinct), deduce that there exists a unique polynomial q(x) of degree at most n such that for . In fact,
(d) Deduce the Lagrange interpolation formula:
for any .
(e) Prove that
where
Suppose now that
For , the preceding result yields the trapezoidal rule for evaluating the definite integral of a polynomial. For , this result yields Simpson’s rule for evaluating the definite integral of a polynomial.
Let V and W be finite-dimensional vector spaces over F, and let and be the isomorphisms between V and V** and W and W**, respectively, as defined in Theorem 2.26. Let be linear, and define . Prove that the diagram depicted in Figure 2.6 commutes (i.e., prove that ). Visit goo.gl/Lkd6XZ for a solution.
Let V be a finite-dimensional vector space with the ordered basis . Prove that , where is defined in Theorem 2.26.
In Exercises 13 through 17, V denotes a finite-dimensional vector space over F. For every subset S of V, define the annihilator of S as
(a) Prove that is a subspace of V*.
(b) If W is a subspace of V and , prove that there exists such that .
(c) Prove that , where is defined as in Theorem 2.26.
(d) For subspaces and , prove that if and only if .
(e) For subspaces and , show that .
Prove that if W is a subspace of V, then . Hint: Extend an ordered basis of W to an ordered basis of V. Let . Prove that is a basis for .
Suppose that W is a finite-dimensional vector space and that is linear. Prove that .
Use Exercises 14 and 15 to deduce that for any .
In Exercises 17 through 20, assume that V and W are finite-dimensional vector spaces. (It can be shown, however, that these exercises are true for all vector spaces V and W.)
Let T be a linear operator on V, and let W be a subspace of V. Prove that W is T-invariant (as defined in the exercises of Section 2.1) if and only if is -invariant.
Let V be a nonzero vector space over a field F, and let S be a basis for V. (By the corollary to Theorem 1.13 (p. 61) in Section 1.7, every vector space has a basis.) Let be the mapping defined by , the restriction of f to S. Prove that is an isomorphism. Hint: Apply Exercise 35 of Section 2.1.
Let V be a nonzero vector space, and let W be a proper subspace of V (i.e., ).
(a) Let and with . Prove that for any scalar a there exists a function such that and for all x in W. Hint: For the infinite-dimensional case, use Exercise 4 of Section 1.7 and Exercise 35 of Section 2.1.
(b) Use (a) to prove there exists a nonzero linear functional such that for all .
Let V and W be nonzero vector spaces over the same field, and let be a linear transformation.
(a) Prove that T is onto if and only if is one-to-one.
(b) Prove that is onto if and only if T is one-to-one.
Hint: In the infinite-dimensional case, use Exercise 19 for parts of the proof.