The concept of invertibility is introduced quite early in the study of functions. Fortunately, many of the intrinsic properties of functions are shared by their inverses. For example, in calculus we learn that the properties of being continuous or differentiable are generally retained by the inverse functions. We see in this section (Theorem 2.17) that the inverse of a linear transformation is also linear. This result greatly aids us in the study of inverses of matrices. As one might expect from Section 2.3, the inverse of the left-multiplication transformation (when it exists) can be used to determine properties of the inverse of the matrix A.
In the remainder of this section, we apply many of the results about in- vertibility to the concept of isomorphism. We will see that finite-dimensional vector spaces (over F) of equal dimension may be identified. These ideas will be made precise shortly.
The facts about inverse functions presented in Appendix B are, of course, true for linear transformations. Nevertheless, we repeat some of the definitions for use in this section.
Let V and W be vector spaces, and let be linear. A function is said to be an inverse of T if and . If T has an inverse, then T is said to be invertible. As noted in Appendix B, if T is invertible, then the inverse of T is unique and is denoted by .
The following facts hold for invertible functions T and U.
in particular, is invertible.
We often use the fact that a function is invertible if and only if it is both one-to-one and onto. We can therefore restate Theorem 2.5 as follows.
Let be a linear transformation, where V and W are finite-dimensional spaces of equal dimension. Then T is invertible if and only if .
Let be the linear transformation defined by . The reader can verify directly that is defined by . Observe that is also linear. As Theorem 2.17 demonstrates, this is true in general.
Let V and W be vector spaces, and let be linear and invertible. Then is linear.
Let and . Since T is onto and one-to-one, there exist unique vectors and such that and . Thus and so
Let T be an invertible linear transformation from V to W. Then V is finite-dimensional if and only if W is finite-dimensional. In this case, .
Suppose that V is finite-dimensional. Let be a basis for V. By Theorem 2.2 (p. 68), spans hence W is finite-dimensional by Theorem 1.9 (p. 45). Conversely, if W is finite-dimensional, then so is V by a similar argument, using .
Now suppose that V and W are finite-dimensional. Because T is one-to-one and onto, we have
So by the dimension theorem (p. 70), it follows that .
It now follows immediately from Theorem 2.5 (p. 71) that if T is a linear transformation between vector spaces of equal (finite) dimension, then the conditions of being invertible, one-to-one, and onto are all equivalent.
We are now ready to define the inverse of a matrix. The reader should note the analogy with the inverse of a linear transformation.
Let A be an matrix. Then A is invertible if there exists an matrix B such that .
If A is invertible, then the matrix B such that is unique. (If C were another such matrix, then .) The matrix B is called the inverse of A and is denoted by .
The reader should verify that the inverse of
In Section 3.2, we will learn a technique for computing the inverse of a matrix. At this point, we develop a number of results that relate the inverses of matrices to the inverses of linear transformations.
Let V and W be finite-dimensional vector spaces with ordered bases and , respectively. Let be linear. Then T is invertible if and only if is invertible. Furthermore, .
Suppose that T is invertible. By the Corollary to Theorem 2.17, we have . Let . So is an matrix. Now satisfies and . Thus
Similarly, . So is invertible and .
where and . It follows that . To show that observe that
Let and be the standard ordered bases of and , respectively. For T as in Example 1, we have
It can be verified by matrix multiplication that each matrix is the inverse of the other.
Let V be a finite-dimensional vector space with an ordered basis , and let be linear. Then T is invertible if and only if is invertible. Furthermore, .
Let A be an matrix. Then A is invertible if and only if is invertible. Furthermore, .
The notion of invertibility may be used to formalize what may already have been observed by the reader, that is, that certain vector spaces strongly resemble one another except for the form of their vectors. For example, in the case of and , if we associate to each matrix
the 4-tuple (a, b, c, d), we see that sums and scalar products associate in a similar manner; that is, in terms of the vector space structure, these two vector spaces may be considered identical or isomorphic.
Let V and W be vector spaces. We say that V is isomorphic to W if there exists a linear transformation that is invertible. Such a linear transformation is called an isomorphism from V onto W.
Define by . It is easily checked that T is an isomorphism; so is isomorphic to .
It is easily verified that T is linear. By use of the Lagrange interpolation formula in Section 1.6, it can be shown (compare with Exercise 22) that only when f is the zero polynomial. Thus T is one-to-one (see Exercise 11). Moreover, because , it follows that T is invertible by Theorem 2.5 (p. 71). We conclude that is isomorphic to .
In each of Examples 4 and 5, the reader may have observed that isomor-phic vector spaces have equal dimensions. As the next theorem shows, this is no coincidence.
Let V and W be finite-dimensional vector spaces (over the same field). Then V is isomorphic to W if and only if .
Suppose that V is isomorphic to W and that is an isomorphism from V to W. By the lemma preceding Theorem 2.18, we have that .
By the lemma to Theorem 2.18, if V and W are isomorphic, then either both of V and W are finite-dimensional or both are infinite-dimensional.
Let V be a vector space over F. Then V is isomorphic to if and only if .
Up to this point, we have associated linear transformations with their matrix representations. We are now in a position to prove that, as a vector space, the collection of all linear transformations between two given vector spaces may be identified with the appropriate vector space of matrices.
Let V and W be finite-dimensional vector spaces over F of dimensions n and m, respectively, and let and be ordered bases for V and W, respectively. Then the function , defined by for , is an isomorphism.
By Theorem 2.8 (p. 83), is linear. Hence we must show that is one-to-one and onto. This is accomplished if we show that for every matrix A, there exists a unique linear transformation such that . Let , and let A be a given matrix. By Theorem 2.6 (p. 73), there exists a unique linear transformation such that
But this means that . So is an isomorphism.
Let V and W be finite-dimensional vector spaces of dimensions n and m, respectively. Then is finite-dimensional of dimension mn.
We conclude this section with a result that allows us to see more clearly the relationship between linear transformations defined on abstract finite- dimensional vector spaces and linear transformations from to .
We begin by naming the transformation introduced in Section 2.2.
Let be an ordered basis for an n-dimensional vector space V over the field F. The standard representation of V with respect to is the function defined by for each .
Let and . It is easily observed that and are ordered bases for . For , we have
We observed earlier that is a linear transformation. The next theorem tells us much more.
For any finite-dimensional vector space V with ordered basis is an isomorphism.
This theorem provides us with an alternate proof that an n-dimensional vector space is isomorphic to (see the corollary to Theorem 2.19).
Let V and W be vector spaces of dimension n and m, respectively, and let be a linear transformation. Define , where and are arbitrary ordered bases of V and W, respectively. We are now able to use and to study the relationship between the linear transformations T and .
Let us first consider Figure 2.2. Notice that there are two composites of linear transformations that map V into :
Map V into with and follow this transformation with this yields the composite .
Map V into W with T and follow it by to obtain the composite .
that is, the diagram “commutes.” Heuristically, this relationship indicates that after V and W are identified with and via and , respectively, we may “identify” T with . This diagram allows us to transfer operations on abstract vector spaces to ones on and .
Recall the linear transformation defined in Example 4 of Section 2.2 . Let and be the standard ordered bases for and , respectively, and let and be the corresponding standard representations of and . If , then
Consider the polynomial . We show that . Now
But since , we have
Try repeating Example 7 with different polynomials p(x).
Label the following statements as true or false. In each part, V and W are vector spaces with ordered (finite) bases and , respectively, is linear, and A and B are matrices.
(b) T is invertible if and only if T is one-to-one and onto.
(d) is isomorphic to .
(e) is isomorphic to if and only if .
(f) implies that A and B are invertible.
(g) If A is invertible, then .
(h) A is invertible if and only if is invertible.
(i) A must be square in order to possess an inverse.
For each of the following linear transformations T, determine whether T is invertible and justify your answer.
(a) defined by .
(b) defined by .
(c) defined by .
(d) defined by .
(e) defined by .
(f) defined by .
Which of the following pairs of vector spaces are isomorphic? Justify your answers.
(a) and .
(b) and .
(c) and .
(d) and .
Let A and B be invertible matrices. Prove that AB is invertible and .
Prove that if A is invertible and , then .
Let A be an matrix.
(a) Suppose that . Prove that A is not invertible.
(b) Suppose that for some nonzero matrix B. Could A be invertible? Explain.
† Let A and B be matrices such that .
(a) Use Exercise 9 to conclude that A and B are invertible.
(b) Prove (and hence ). (We are, in effect, saying that for square matrices, a “one-sided” inverse is a “two-sided” inverse.)
(c) State and prove analogous results for linear transformations defined on finite-dimensional vector spaces.
Verify that the transformation in Example 5 is one-to-one.
Prove Theorem 2.21.
Let mean “is isomorphic to.” Prove that is an equivalence relation on the class of vector spaces over F.
Construct an isomorphism from V to .
Let V and W be n-dimensional vector spaces, and let be a linear transformation. Suppose that is a basis for V. Prove that T is an isomorphism if and only if is a basis for W.
Let B be an invertible matrix. Define by . Prove that is an isomorphism.
† Let V and W be finite-dimensional vector spaces and be an isomorphism. Let be a subspace of V.
(a) Prove that is a subspace of W.
(b) Prove that .
Repeat Example 7 with the polynomial .
First prove that is a basis for . Then let be the matrix with 1 in the ith row and jth column and 0 elsewhere, and prove that . Again by Theorem 2.6, there exists a linear transformation such that . Prove that is an isomorphism.
Let be distinct scalars from an infinite field F. Define by . Prove that T is an isomorphism. Hint: Use the Lagrange polynomials associated with
Let V and Z be vector spaces and be a linear transformation that is onto. Define the mapping
for any coset in .
(a) Prove that is well-defined; that is, prove that if , then .
(b) Prove that is linear.
(c) Prove that is an isomorphism.
(d) Prove that the diagram shown in Figure 2.3 commutes; that is, prove that .
Let V be a nonzero vector space over a field F, and suppose that S is a basis for V. (By the corollary to Theorem 1.13 (p. 61) in Section 1.7, every vector space has a basis.) Let C(S, F) denote the vector space of all functions such that for all but a finite number of vectors in S. (See Exercise 14 of Section 1.3.) Let be defined by if f is the zero function, and
otherwise. Prove that is an isomorphism. Thus every nonzero vector space can be viewed as a space of functions.