Linear algebra friedberg pdf free download

Linear algebra friedberg pdf free download

Linear Algebra - Friedberg,linear algebra 4th edition stephen h friedberg pdf free download

Download Linear Algebra - Friedberg; Insel; Spence [4th E] Type: PDF. Date: July Size: MB. Author: Luis Roberto Ríos. This document was uploaded by user and they Linear Algebra Friedberg 5Th Edition PDF Download. The book, Linear Algebra Friedberg 5Th Edition, has been published. The information in this book is intended for those who are [PDF] Linear Algebra - Friedberg; Insel; Spence [4th E] - Free Download PDF DLSCRIB - Free, Fast and Secure Home Linear Algebra - Friedberg; Insel; Spence [4th E] Linear Algebra - You can read this full article here: friedberg linear algebra pdf. Friedberg Linear Algebra by Louis Antoine Wolf, Gregory Kontorovich, David Eisenbud is a book written for graduate Linear Algebra, 4th edition Linear Algebra This page intentionally left blank Fourth Edition Stephen H. Friedberg Arnold J. Insel Lawrence E. S. 26, 3, 3MB. Pages Page size ... read more

friedberg linear algebra pdf This Friedberg Linear Algebra PDF ebook is meant to be easy to read and understand, so you can use it for your reference. Linear Algebra. Know More. Linear Algebra 2-downloads. LINEAR ALGEBRA, 4TH EDN [Paperback] Friedberg, Insel, Spence. Linear Algebra, Books a la Carte edition. Leave a Comment Cancel reply Comment Name Email Website Save my name, email, and website in this browser for the next time I comment. A subset S of a vector space that is not linearly dependent is called linearly independent. As before, we also say that the vectors of S are linearly independent. The following facts about linearly independent sets are true in any vector space. The empty set is linearly independent, for linearly dependent sets must be nonempty. A set consisting of a single nonzero vector is linearly independent. A set is linearly independent if and only if the only representations of 0 as linear combinations of its vectors are trivial representations.

This technique is illustrated in the examples that follow. Equating the corresponding coordinates of the vectors on the left and the right sides of this equation, we obtain the following system of linear equations. If S1 is linearly dependent, then S2 is linearly dependent. If S2 is linearly independent, then S1 is linearly independent. Earlier in this section, we remarked that the issue of whether S is the smallest generating set for its span is related to the question of whether some vector in S is a linear combination of the other vectors in S. Thus the issue of whether S is the smallest generating set for its span is related to the question of whether S is linearly dependent.

This equation implies that u3 or alternatively, u1 or u2 is a linear combination of the other vectors in S. More generally, suppose that S is any linearly dependent set containing two or more vectors. It follows that if no proper subset of S generates the span of S, then S must be linearly independent. Another way to view the preceding statement is given in Theorem 1. Let S be a linearly independent subset of a vector space V, and let v be a vector in V that is not in S. Since v is a linear combination of u2 ,. Then there exist vectors v1 , v2 ,. Linearly independent generating sets are investigated in detail in Section 1.

a If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S. b Any set containing the zero vector is linearly dependent. c The empty set is linearly dependent. d Subsets of linearly dependent sets are linearly dependent. e Subsets of linearly independent sets are linearly independent. Prove that {e1 , e2 , · · · , en } is linearly independent. Show that the set {1, x, x2 ,. In Mm×n F , let E ij denote the matrix whose only nonzero entry is 1 in the ith row and jth column. Recall from Example 3 in Section 1. Find a linearly independent set that generates this subspace. b Prove that if F has characteristic 2, then S is linearly dependent. Show that {u, v} is linearly dependent if and only if u or v is a multiple of the other. Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another.

How many vectors are there in span S? Prove Theorem 1. Let V be a vector space over a field of characteristic not equal to two. a Let u and v be distinct vectors in V. b Let u, v, and w be distinct vectors in V. A linearly independent generating set for W possesses a very useful property—every vector in W can be expressed in one and only one way as a linear combination of the vectors in the set. This property is proved below in Theorem 1. It is this property that makes linearly independent generating sets the building blocks of vector spaces. A basis β for a vector space V is a linearly independent subset of V that generates V. If β is a basis for V, we also say that the vectors of β form a basis for V. We call this basis the standard basis for Pn F.

In fact, later in this section it is shown that no basis for P F can be finite. Hence not every vector space has a finite basis. The next theorem, which is used frequently in Chapter 2, establishes the most significant property of a basis. Let β be a basis for V. Thus v is a linear combination of the vectors of β. The proof of the converse is an exercise. Thus v determines a unique n-tuple of scalars a1 , a2 ,. This fact suggests that V is like the vector space Fn , where n is the number of vectors in the basis for V. We see in Section 2.

In this book, we are primarily interested in vector spaces having finite bases. If a vector space V is generated by a finite set S, then some subset of S is a basis for V. Hence V has a finite basis. Otherwise S contains a nonzero vector u1. By item 2 on page 37, {u1 } is a linearly independent set. Continue, if possible, choosing vectors u2 ,. We claim that β is a basis for V. Because β is linearly independent by construction, it suffices to show that β spans V. By Theorem 1. Because of the method by which the basis β was obtained in the proof of Theorem 1. This method is illustrated in the next example. It can be shown that S generates R3.

We can select a basis for R3 that is a subset of S by the technique used in proving Theorem 1. Let V be a vector space that is generated by a set G containing exactly n vectors, and let L be a linearly independent subset of V containing exactly m vectors. The proof is by mathematical induction on m. By the corollary to Theorem 1. Thus there exist scalars a1 , a2 ,. Moreover, some bi , say b1 , is nonzero, for otherwise we obtain the same contradiction. Because {v1 , v2 ,. This completes the induction. Let V be a vector space having a finite basis. Then every basis for V contains the same number of vectors. Suppose that β is a finite basis for V that contains exactly n vectors, and let γ be any other basis for V. If a vector space has a finite basis, Corollary 1 asserts that the number of vectors in any basis for V is an intrinsic property of V. This fact makes possible the following important definitions.

A vector space is called finite-dimensional if it has a basis consisting of a finite number of vectors. The unique number of vectors Sec. A vector space that is not finite-dimensional is called infinite-dimensional. The following results are consequences of Examples 1 through 4. Example 7 The vector space {0 } has dimension zero. Example 11 Over the field of complex numbers, the vector space of complex numbers has dimension 1. A basis is {1}. A basis is {1, i}. From this fact it follows that the vector space P F is infinite-dimensional because it has an infinite linearly independent set, namely {1, x, x2 ,.

This set is, in fact, a basis for P F. Yet nothing that we have proved in this section guarantees an infinite-dimensional vector space must have a basis. In Section 1. Just as no linearly independent subset of a finite-dimensional vector space V can contain more than dim V vectors, a corresponding statement can be made about the size of a generating set. Let V be a vector space with dimension n. a Any finite generating set for V contains at least n vectors, and a generating set for V that contains exactly n vectors is a basis for V.

c Every linearly independent subset of V can be extended to a basis for V. a Let G be a finite generating set for V. Corollary 1 implies that H contains exactly n vectors. Since a subset of G contains n vectors, G must contain at least n vectors. b Let L be a linearly independent subset of V containing exactly n vectors. Since L is also linearly independent, L is a basis for V. Example 13 It follows from Example 4 of Section 1. It follows from Example 4 of Section 1. This procedure also can be used to extend a linearly independent set to a basis, as c of Corollary 2 asserts is possible.

An Overview of Dimension and Its Consequences Theorem 1. For this reason, we summarize here the main results of this section in order to put them into better perspective. A basis for a vector space V is a linearly independent subset of V that generates V. If V has a finite basis, then every basis for V contains the same number of vectors. This number is called the dimension of V, and V is said to be finite-dimensional. Thus if the dimension of V is n, every basis for V contains exactly n vectors. Moreover, every linearly independent subset of V contains no more than n vectors and can be extended to a basis for V by including appropriately chosen vectors. Also, each generating set for V contains at least n vectors and can be reduced to a basis for V by excluding appropriately chosen vectors. The Venn diagram in Figure 1. Linearly independent Bases sets Figure 1. Let W be a subspace of a finite-dimensional vector space V. Otherwise, W contains a nonzero vector x1 ; so {x1 } is a linearly independent set.

Continue choosing vectors, x1 , x2 ,. If W is a subspace of a finite-dimensional vector space V, then any basis for W can be extended to a basis for V. Let S be a basis for W. Because S is a linearly independent subset of V, Corollary 2 of the replacement theorem guarantees that S can be extended to a basis for V. A basis for W is {1, x2 ,. Since R2 has dimension 2, subspaces of R2 can be of dimensions 0, 1, or 2 only. The only subspaces of dimension 0 or 2 are {0 } and R2 , respectively. Any subspace of R2 having dimension 1 consists of all scalar multiples of some nonzero vector in R2 Exercise 11 of Section 1. If a point of R2 is identified in the natural way with a point in the Euclidean plane, then it is possible to describe the subspaces of R2 geometrically: A subspace of R2 having dimension 0 consists of the origin of the Euclidean plane, a subspace of R2 with dimension 1 consists of a line through the origin, and a subspace of R2 having dimension 2 is the entire Euclidean plane.

Similarly, the subspaces of R3 must have dimensions 0, 1, 2, or 3. Interpreting these possibilities geometrically, we see that a subspace of dimension zero must be the origin of Euclidean 3-space, a subspace of dimension 1 is a line through the origin, a subspace of dimension 2 is a plane through the origin, and a subspace of dimension 3 is Euclidean 3-space itself. The Lagrange Interpolation Formula Corollary 2 of the replacement theorem can be applied to obtain a useful formula. The polynomials f0 x , f1 x ,. Note that each fi x is a polynomial of degree n and hence is in Pn F.

This representation is called the Lagrange interpolation formula. Notice Sec. a b c d The zero vector space has no basis. Every vector space that is generated by a finite set has a basis. Every vector space has a finite basis. A vector space cannot have more than one basis. f The dimension of Pn F is n. h Suppose that V is a finite-dimensional vector space, that S1 is a linearly independent subset of V, and that S2 is a subset of V that generates V. Then S1 cannot contain more vectors than S2. i If S generates the vector space V, then every vector in V can be written as a linear combination of vectors in S in only one way. j Every subspace of a finite-dimensional space is finite-dimensional. k If V is a vector space having dimension n, then V has exactly one subspace with dimension 0 and exactly one subspace with dimension n.

l If V is a vector space having dimension n, and if S is a subset of V with n vectors, then S is linearly independent if and only if S spans V. Determine which of the following sets are bases for R3. Determine which of the following sets are bases for P2 R. Give three different bases for F2 and for M2×2 F. Find a subset of the set {u1 , u2 , u3 , u4 , u5 } that is a basis for R3. Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. Find a subset of the set {u1 , u2 ,. Find the unique representation of an arbitrary vector a1 , a2 , a3 , a4 in F4 as a linear combination of u1 , u2 , u3 , and u4. In each part, use the Lagrange interpolation formula to construct the polynomial of smallest degree whose graph contains the following points. Let u and v be distinct vectors of a vector space V.

Let u, v, and w be distinct vectors of a vector space V. Find a basis for this subspace. What are the dimensions of W1 and W2? The set of all n × n matrices having trace equal to zero is a subspace W of Mn×n F see Example 4 of Section 1. Find a basis for W. What is the dimension of W? The set of all upper triangular n × n matrices is a subspace W of Mn×n F see Exercise 12 of Section 1. The set of all skew-symmetric n × n matrices is a subspace W of Mn×n F see Exercise 28 of Section 1. Find a basis for the vector space in Example 5 of Section 1. Complete the proof of Theorem 1. a Prove that there is a subset of S that is a basis for V. Be careful not to assume that S is finite. b Prove that S contains at least n vectors. Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset.

Let W1 and W2 be subspaces of a finite-dimensional vector space V. Let f x be a polynomial of degree n in Pn R. Let V, W, and Z be as in Exercise 21 of Section 1. If V and W are vector spaces over F of dimensions m and n, determine the dimension of Z. Let W1 and W2 be the subspaces of P F defined in Exercise 25 in Section 1. Let V be a finite-dimensional vector space over C with dimension n. See Examples 11 and Exercises 29—34 require knowledge of the sum and direct sum of subspaces, as defined in the exercises of Section 1. Hint: Start with a basis {u1 , u2 ,. vm } for W1 and to a basis {u1 , u2 ,. wp } for W2. Our principal goal here is to prove that every vector space has a basis. This result is important in the study of infinite-dimensional vector spaces because it is often difficult to construct an explicit basis for such a space.

Consider, for example, the vector space of real numbers over the field of rational numbers. There is no obvious way to construct a basis for this space, and yet it follows from the results of this section that such a basis does exist. The difficulty that arises in extending the theorems of the preceding section to infinite-dimensional vector spaces is that the principle of mathematical induction, which played a crucial role in many of the proofs of Section 1. Instead, a more general result called the maximal principle is needed. Before stating this principle, we need to introduce some terminology. Let F be a family of sets. A member M of F is called maximal with respect to set inclusion if M is contained in no member of F other than M itself. This family F is called the power set of S. The set S is easily seen to be a maximal element of F. Then S and T are both maximal elements of F.

Then F has no maximal element. Maximal Principle. Because the maximal principle guarantees the existence of maximal elements in a family of sets satisfying the hypothesis above, it is useful to reformulate the definition of a basis in terms of a maximal property. In Theorem 1. Let S be a subset of a vector space V. A maximal linearly independent subset of S is a subset B of S satisfying both of the following conditions. a B is linearly independent. b The only linearly independent subset of S that contains B is B itself. For a treatment of set theory using the Maximal Principle, see John L. Kelley, General Topology, Graduate Texts in Mathematics Series, Vol. In this case, however, any subset of S consisting of two polynomials is easily shown to be a maximal linearly independent subset of S. Thus maximal linearly independent subsets of a set need not be unique. β is linearly independent by definition.

Our next result shows that the converse of this statement is also true. Let V be a vector space and S a subset that generates V. If β is a maximal linearly independent subset of S, then β is a basis for V. Let β be a maximal linearly independent subset of S. Because β is linearly independent, it suffices to prove that β generates V. Since Theorem 1. Thus a subset of a vector space is a basis if and only if it is a maximal linearly independent subset of the vector space. Therefore we can accomplish our goal of proving that every vector space has a basis by showing that every vector space contains a maximal linearly independent subset. This result follows immediately from the next theorem. Let S be a linearly independent subset of a vector space V.

There exists a maximal linearly independent subset of V that contains S. Let F denote the family of all linearly independent subsets of V that contain S. In order to show that F contains a maximal element, we must show that if C is a chain in F, then there exists a member U of F that contains each member of C. We claim that U , the union of the members of C, is the desired set. Clearly U contains each member of C, and so it suffices to prove Sec. Thus we need only prove that U is linearly independent.

But since C is a chain, one of these sets, say Ak , contains all the others. It follows that U is linearly independent. The maximal principle implies that F has a maximal element. This element is easily seen to be a maximal linearly independent subset of V that contains S. Every vector space has a basis. It can be shown, analogously to Corollary 1 of the replacement theorem p. Sets have the same cardinality if there is a one-to-one and onto mapping between them. See, for example, N. Jacobson, Lectures in Abstract Algebra, vol. Van Nostrand Company, New York, , p. Exercises extend other results from Section 1. a Every family of sets contains a maximal element. b Every chain contains a maximal element. c If a family of sets has a maximal element, then that maximal element is unique. d If a chain of sets has a maximal element, then that maximal element is unique. e A basis for a vector space is a maximal linearly independent subset of that vector space.

f A maximal linearly independent subset of a vector space is a basis for that vector space. Show that the set of convergent sequences is an infinite-dimensional subspace of the vector space of all sequences of real numbers. See Exercise 21 in Section 1. Let V be the set of real numbers regarded as a vector space over the field of rational numbers. Prove that V is infinite-dimensional. Hint: 62 Chap. Let W be a subspace of a not necessarily finite-dimensional vector space V. Prove that any basis for W is a subset of a basis for V. Prove the following infinite-dimensional version of Theorem 1. Then β is a basis for V if and only if for each nonzero vector v in V, there exist unique vectors u1 , u2 ,. Prove the following generalization of Theorem 1. Hint: Apply the maximal principle to the family of all linearly independent subsets of S2 that contain S1 , and proceed as in the proof of Theorem 1. Prove the following generalization of the replacement theorem. Let β be a basis for a vector space V, and let S be a linearly independent subset of V.

These special functions are called linear transformations, and they abound in both pure and applied mathematics. In calculus, the operations of differentiation and integration provide us with two of the most important examples of linear transformations see Examples 6 and 7 of Section 2. These two examples allow us to reformulate many of the problems in differential and integral equations in terms of linear transformations on particular vector spaces see Sections 2. In geometry, rotations, reflections, and projections see Examples 2, 3, and 4 of Section 2.

Later we use these transformations to study rigid motions in Rn Section 6. In the remaining chapters, we see further examples of linear transformations occurring in both the physical and the social sciences. Throughout this chapter, we assume that all vector spaces are over a common field F. Many of these transformations are studied in more detail in later sections. Recall that a function T with domain V and codomain W is denoted by 64 Sec. See Appendix B. Let V and W be vector spaces over F. If the underlying field F is the field of rational numbers, then a implies b see Exercise 37 , but, in general a and b are logically independent. See Exercises 38 and We often simply call T linear.

See Exercise 7. T is linear if and only if, for x1 , x2 ,. So T is linear. The main reason for this is that most of the important geometrical transformations are linear. Three particular transformations that we now consider are rotation, reflection, and projection. We leave the proofs of linearity to the reader. We determine an explicit formula for Tθ. It is now easy to show, as in Example 1, that Tθ is linear. T is called the reflection about the x -axis. See Figure 2. T is called the projection on the x -axis. Then T is a linear transformation by Exercise 3 of Section 1. Then T is a linear transformation because the definite integral of a linear combination of functions is the same as the linear combination of the definite integrals of the functions. It is clear that both of these transformations are linear. We often write I instead of IV. We now turn our attention to two very important sets associated with linear transformations: the range and null space.

The determination of these sets allows us to examine more closely the intrinsic properties of a linear transformation. The next result shows that this is true in general. Theorem 2. Then N T and R T are subspaces of V and W, respectively. To clarify the notation, we use the symbols 0 V and 0 W to denote the zero vectors of V and W, respectively. The next theorem provides a method for finding a spanning set for the range of a linear transformation. With this accomplished, a basis for the range is easy to discover using the technique of Example 6 of Section 1. Because R T is a subspace, R T contains span {T v1 , T v2 ,. It should be noted that Theorem 2. The next example illustrates the usefulness of Theorem 2.

The null space and range are so important that we attach special names to their respective dimensions. If N T and R T are finite-dimensional, then we define the nullity of T, denoted nullity T , and the rank of T, denoted rank T , to be the dimensions of N T and R T , respectively. Reflecting on the action of a linear transformation, we see intuitively that the larger the nullity, the smaller the rank. In other words, the more vectors that are carried into 0 , the smaller the range. The same heuristic reasoning tells us that the larger the rank, the smaller the nullity. This balance between rank and nullity is made precise in the next theorem, appropriately called the dimension theorem. First we prove that S generates R T. Using Theorem 2. Now we prove that S is linearly independent. Hence S is linearly independent. Interestingly, for a linear transformation, both of these concepts are intimately connected to the rank and nullity of the transformation.

This is demonstrated in the next two theorems. This means that T is one-to-one. The reader should observe that Theorem 2. Surprisingly, the conditions of one-to-one and onto are equivalent in an important special case. Then the following are equivalent. a T is one-to-one. b T is onto. Now, with the use of Theorem 2. See Exercises 15, 16, and The linearity of T in Theorems 2. The next two examples make use of the preceding theorems in determining whether a given linear transformation is one-to-one or onto. We conclude from Theorem 2. Hence Theorem 2. Example 13 illustrates the use of this result. Clearly T is linear and one-to-one. This technique is exploited more fully later. One of the most important properties of a linear transformation is that it is completely determined by its action on a basis.

This result, which follows from the next theorem and corollary, is used frequently throughout the book. Let V and W be vector spaces over F , and suppose that {v1 , v2 ,. Let V and W be vector spaces, and suppose that V has a finite basis {v1 , v2 ,. This follows from the corollary and from the fact that { 1, 2 , 1, 1 } is a basis for R2. In each part, V and W are finite-dimensional vector spaces over F , and T is a function from V to W. a If T is linear, then T preserves sums and scalar products. f If T is linear, then T carries linearly independent subsets of V onto linearly independent subsets of W.

For Exercises 2 through 6, prove that T is a linear transformation, and find bases for both N T and R T. Then compute the nullity and rank of T, and verify the dimension theorem. Finally, use the appropriate theorems in this section to determine whether T is one-to-one or onto. Recall Example 4, Section 1. Prove properties 1, 2, 3, and 4 on page Prove that the transformations in Examples 2 and 3 are linear. For each of the following parts, state why T is not linear. What is T 2, 3? Is T one-to-one? What is T 8, 11? a Prove that T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. b Suppose that T is one-to-one and that S is a subset of V. Prove that S is linearly independent if and only if T S is linearly independent.

Recall the definition of P R on page Recall that T is linear. Prove that T is onto, but not one-to-one. a Prove that if dim V dim W , then T cannot be one-to-one. Let V and W be vector spaces with subspaces V1 and W1 , respectively. Let V be the vector space of sequences described in Example 5 of Section 1. T and U are called the left shift and right shift operators on V, respectively. a Prove that T and U are linear. b Prove that T is onto, but not one-to-one. c Prove that U is one-to-one, but not onto. Describe geometrically the possibilities for the null space of T. Hint: Use Exercise The following definition is used in Exercises 24—27 and in Exercise Recall the definition of direct sum given in the exercises of Section 1. Include figures for each of the following parts. b Find a formula for T a, b, c , where T represents the projection on the z-axis along the xy-plane.

Describe T if W1 is the zero subspace. Suppose that W is a subspace of a finite-dimensional vector space V. b Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces. The following definitions are used in Exercises 28— Warning: Do not assume that W is T-invariant or that T is a projection unless explicitly stated. Prove that the subspaces {0 }, V, R T , and N T are all T-invariant. If W is T-invariant, prove that TW is linear. c Show by example that the conclusion of b is not necessarily true if V is not finite-dimensional. Suppose that W is T-invariant. Prove Theorem 2. Prove the following generalization of Theorem 2. Exercises 35 and 36 assume the definition of direct sum given in the exercises of Section 1.

Be careful to say in each part where finite-dimensionality is used. Let V and T be as defined in Exercise Thus the result of Exercise 35 a above cannot be proved without assuming that V is finite-dimensional. Conclude that V being finite-dimensional is also essential in Exercise 35 b. Prove that if V and W are vector spaces over the field of rational numbers, then any additive function from V into W is a linear transformation. Prove that T is additive as defined in Exercise 37 but not linear. Hint: Let V be the set of real numbers regarded as a vector space over the field of rational numbers. By Exercise 34, there exists a linear transformation Sec. The following exercise requires familiarity with the definition of quotient space given in Exercise 31 of Section 1.

Let V be a vector space and W be a subspace of V. b Suppose that V is finite-dimensional. c Read the proof of the dimension theorem. Compare the method of solving b with the method of deriving the same result as outlined in Exercise 35 of Section 1. In this section, we embark on one of the most useful approaches to the analysis of a linear transformation on a finite-dimensional vector space: the representation of a linear transformation by a matrix. In fact, we develop a one-to-one correspondence between matrices and linear transformations that allows us to utilize properties of one to study properties of the other.

We first need the concept of an ordered basis for a vector space. Let V be a finite-dimensional vector space. An ordered basis for V is a basis for V endowed with a specific order; that is, an ordered basis for V is a finite sequence of linearly independent vectors in V that generates V. Similarly, for the vector space Pn F , we call {1, x,. Now that we have the concept of ordered basis, we can identify abstract vectors in an n-dimensional vector space with n-tuples. This identification is provided through the use of coordinate vectors, as introduced next. We study this transformation in Section 2. Notice that the jth column of A is simply [T vj ]γ. We illustrate the computation of [T]γβ in the next several examples.

Let β and γ be the standard ordered bases for R2 and R3 , respectively. Let β and γ be the standard ordered bases for P3 R and P2 R , respectively. To make this more explicit, we need some preliminary discussion about the addition and scalar multiplication of linear transformations. Of course, these are just the usual definitions of addition and scalar multiplication of functions. We are fortunate, however, to have the result that both sums and scalar multiples of linear transformations are also linear. b Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations from V to W is a vector space over F. b Noting that T0 , the zero transformation, plays the role of the zero vector, it is easy to verify that the axioms of a vector space are satisfied, and hence that the collection of all linear transformations from V into W is a vector space over F.

We denote the vector space of all linear transformations from V into W by L V, W. In Section 2. This identification is easily established by the use of the next theorem. Then Sec. So a is proved, and the proof of b is similar. Let β and γ be the standard ordered bases of R2 and R3 , respectively. L V, W is a vector space. Let β and γ be the standard ordered bases for Rn and Rm , respectively. Compute [T]γβ. Compute [T]α. Compute [T]α β. Compute [T]γα. Complete the proof of part b of Theorem 2. Prove part b of Theorem 2. Prove that T is linear. Let V be the vector space of complex numbers over the field R. Recall by Exercise 38 of Section 2. By Theorem 2. Compute [T]β. Suppose that W is a T-invariant subspace of V see the exercises of Section 2. See the definition in the exercises of Section 2. Find an ordered basis β for V such that [T]β is a diagonal matrix.

Let V and W be vector spaces, and let T and U be nonzero linear transformations from V into W. Prove that the set {T1 , T2 ,. Let V and W be vector spaces, and let S be a subset of V. Prove the following statements. Show that there exist ordered bases β and γ for V and W, respectively, such that [T]γβ is a diagonal matrix. The question now arises as to how the matrix representation of a composite of linear transformations is related to the matrix representation of each of the associated linear transformations. The attempt to answer this question leads to a definition of matrix multiplication. Our first result shows that the composite of linear transformations is linear. Let V be a vector space. A more general result holds for linear transformations that have domains unequal to their codomains. See Exercise 8. Consider the matrix [UT]γα. Some interesting applications of this definition are presented at the end of this section.

Recalling the definition of the transpose of a matrix from Section 1. Therefore the transpose of a product is the product of the transposes in the opposite order. The next theorem is an immediate consequence of our definition of matrix multiplication. Let V, W, and Z be finite-dimensional vector spaces with ordered bases α, β, and γ, respectively. Let V be a finite-dimensional vector space with an ordered basis β. We illustrate Theorem 2. To illustrate Theorem 2. Observe also that part c of the next theorem illustrates that the identity matrix acts as a multiplicative identity in Mn×n F. When the context is clear, we sometimes omit the subscript n from In. Let A be an m × n matrix, B and C be n × p matrices, and D and E be q × m matrices. We prove the first half of a and c and leave the remaining proofs as an exercise. See Exercise 5. Thus the cancellation property for multiplication in fields is not valid for matrices.

To see why, assume that the cancellation law is valid. The proof of b is left as an exercise. See Exercise 6. It follows see Exercise 14 from Theorem 2. An analogous result holds for rows; that is, row i of AB is a linear combination of the rows of B with the coefficients in the linear combination being the entries of row i of A. The next result justifies much of our past work. It utilizes both the matrix representation of a linear transformation and matrix multiplication in order to evaluate the transformation at any given vector. Identifying column vectors as matrices and using Theorem 2. This transformation is probably the most important tool for transferring properties about transformations to analogous properties about matrices and vice versa. For example, we use it to prove that matrix multiplication is associative. Let A be an m × n matrix with entries from a field F.

We call LA a left-multiplication transformation. These properties are all quite natural and so are easy to remember. Let A be an m × n matrix with entries from F. Furthermore, if B is any other m × n matrix with entries from F and β and γ are the standard ordered bases for Fn and Fm , respectively, then we have the following properties. The fact that LA is linear follows immediately from Theorem 2. a The jth column of [LA ]γβ is equal to LA ej. The proof of the converse is trivial. c The proof is left as an exercise. The uniqueness of C follows from b. f The proof is left as an exercise. We now use left-multiplication transformations to establish the associativity of matrix multiplication. Let A, B, and C be matrices such that A BC is defined. It is left to the reader to show that AB C is defined. Using e of Theorem 2. So from b of Theorem 2. The proof above, however, provides a prototype of many of the arguments that utilize the relationships between linear transformations and matrices.

Applications A large and varied collection of interesting applications arises in connection with special matrices called incidence matrices. An incidence matrix is a square matrix in which all the entries are either zero or one and, for convenience, all the diagonal entries are zero. If we have a relationship on a set of n objects that we denote by 1, 2,. To make things concrete, suppose that we have four people, each of whom owns a communication device. We obtain an interesting interpretation of the entries of A2. Note that any term A3k Ak1 equals 1 if and only if both A3k and Ak1 equal 1, that is, if and only if 3 can send to k and k can send to 1. Thus A2 31 gives the number of ways in which 3 can send to 1 in two stages or in one relay. A maximal collection of three or more people with the property that any two can send to each other is called a clique.

The problem of determining cliques is difficult, but there is a simple method for determining if someone Sec. Our final example of the use of incidence matrices is concerned with the concept of dominance. In other words, there is at least one person who dominates [is dominated by] all others in one or two stages. In fact, it can be shown that any person who dominates [is dominated by] the greatest number of people in the first stage has this property. Compute At , At B, BC t , CB, and CA. Let β and γ be the standard ordered bases of P2 R and R3 , respectively. Then use Theorem 2.

Compute [h x ]β and [U h x ]γ. Then use [U]γβ from a and Theorem 2. For each of the following parts, let T be the linear transformation defined in the corresponding part of Exercise 5 of Section 2. Use Theorem 2. Complete the proof of Theorem 2. Prove b of Theorem 2. Prove c and f of Theorem 2. Now state and prove a more general result involving linear transformations with domains unequal to their codomains. Let A be an n × n matrix. a Prove that if UT is one-to-one, then T is one-to-one. Must U also be one-to-one? b Prove that if UT is onto, then U is onto. Must T also be onto? c Prove that if U and T are one-to-one and onto, then UT is also. Let A and B be n × n matrices. Assume the notation in Theorem 2. a Suppose that z is a column vector in Fp.

Pages Page size DOWNLOAD FILE. Graduate Texts in Mathematics Linear Algebra Volume 23 Springer Graduate Texts in Mathematics 23 Editorial Bo. Linear Algebra This page intentionally left blank Fourth Edition Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Illinois State University PEARSON EDUCATION, Upper Saddle River, New Jersey Library of Congress Cataloging-in-Publication Data Friedberg, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Includes indexes. ISBN 1. Algebra, Linear. Insel, Arnold J. Spence, Lawrence E. Dreifachgewebe: Baumwolle und Kunstseide, schwarz, weiß, Orange × cm. Foto: Gunter Lepkowski, Berlin. Bauhaus-Archiv, Berlin, Inv. c , , , by Pearson Education, Inc.

Upper Saddle River, New Jersey All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 ISBN Pearson Pearson Pearson Pearson Pearson Pearson Pearson Pearson Education, Ltd. Limited, Sydney Education Singapore, Pte. Education -- Japan, Tokyo Education Malaysia, Pte. Ltd To our families: Ruth Ann, Rachel, Jessica, and Jeremy Barbara, Thomas, and Sara Linda, Stephen, and Alison This page intentionally left blank Contents Preface ix 1 Vector Spaces 1 1.

Vector Spaces. Linear Combinations and Systems of Linear Equations. Linear Dependence and Linear Independence. Bases and Dimension. Maximal Linearly Independent Subsets. Index of Definitions. The Matrix Representation of a Linear Transformation Composition of Linear Transformations and Matrix Multiplication. Invertibility and Isomorphisms. The Change of Coordinate Matrix. Dual Spaces. Homogeneous Linear Differential Equations with Constant Coefficients. v vi Table of Contents 3. Systems of Linear Equations—Theoretical Aspects. Systems of Linear Equations—Computational Aspects Index of Definitions. Determinants of Order n. Properties of Determinants. Summary—Important Facts about Determinants A Characterization of the Determinant. Matrix Limits and Markov Chains. Invariant Subspaces and the Cayley—Hamilton Theorem Index of Definitions. The Gram—Schmidt Orthogonalization Process and Orthogonal Complements.

Table of Contents vii 7 Canonical Forms 7. The Jordan Canonical Form II The Minimal Polynomial. The Rational Canonical Form. Appendices A B C D E Sets. Complex Numbers Polynomials. Answers to Selected Exercises. In addition, linear algebra continues to be of great importance in modern treatments of geometry and analysis. The primary purpose of this fourth edition of Linear Algebra is to present a careful treatment of the principal topics of linear algebra and to illustrate the power of the subject through a variety of applications. Our major thrust emphasizes the symbiotic relationship between linear transformations and matrices. However, where appropriate, theorems are stated in the more general infinite-dimensional case.

For example, this theory is applied to finding solutions to a homogeneous linear differential equation and the best approximation by a trigonometric polynomial to a continuous function. Although the only formal prerequisite for this book is a one-year course in calculus, it requires the mathematical sophistication of typical junior and senior mathematics majors. This book is especially suited for a second course in linear algebra that emphasizes abstract vector spaces, although it can be used in a first course with a strong theoretical emphasis. The book is organized to permit a number of different courses ranging from three to eight semester hours in length to be taught from it. The core material vector spaces, linear transformations and matrices, systems of linear equations, determinants, diagonalization, and inner product spaces is found in Chapters 1 through 5 and Sections 6.

Chapters 6 and 7, on inner product spaces and canonical forms, are completely independent and may be studied in either order. In addition, throughout the book are applications to such areas as differential equations, economics, geometry, and physics. These applications are not central to the mathematical development, however, and may be excluded at the discretion of the instructor. We have attempted to make it possible for many of the important topics of linear algebra to be covered in a one-semester course. This goal has led us to develop the major topics with fewer preliminaries than in a traditional approach.

Our treatment of the Jordan canonical form, for instance, does not require any theory of polynomials. The resulting economy permits us to cover the core material of the book omitting many of the optional sections and a detailed discussion of determinants in a one-semester four-hour course for students who have had some prior exposure to linear algebra. Chapter 1 of the book presents the basic theory of vector spaces: subspaces, linear combinations, linear dependence and independence, bases, and dimension. The chapter concludes with an optional section in which we prove ix x Preface that every infinite-dimensional vector space has a basis. Linear transformations and their relationship to matrices are the subject of Chapter 2. We discuss the null space and range of a linear transformation, matrix representations of a linear transformation, isomorphisms, and change of coordinates. Optional sections on dual spaces and homogeneous linear differential equations end the chapter. The application of vector space theory and linear transformations to systems of linear equations is found in Chapter 3.

We have chosen to defer this important subject so that it can be presented as a consequence of the preceding material. This approach allows the familiar topic of linear systems to illuminate the abstract theory and permits us to avoid messy matrix computations in the presentation of Chapters 1 and 2. There are occasional examples in these chapters, however, where we solve systems of linear equations. Of course, these examples are not a part of the theoretical development. The necessary background is contained in Section 1. Determinants, the subject of Chapter 4, are of much less importance than they once were. In a short course less than one year , we prefer to treat determinants lightly so that more time may be devoted to the material in Chapters 5 through 7.

Consequently we have presented two alternatives in Chapter 4—a complete development of the theory Sections 4. Optional Section 4. Chapter 5 discusses eigenvalues, eigenvectors, and diagonalization. One of the most important applications of this material occurs in computing matrix limits. We have therefore included an optional section on matrix limits and Markov chains in this chapter even though the most general statement of some of the results requires a knowledge of the Jordan canonical form. Section 5. Inner product spaces are the subject of Chapter 6. The basic mathematical theory inner products; the Gram—Schmidt process; orthogonal complements; the adjoint of an operator; normal, self-adjoint, orthogonal and unitary operators; orthogonal projections; and the spectral theorem is contained in Sections 6.

Sections 6. Canonical forms are treated in Chapter 7. Sections 7. There are five appendices. The first four, which discuss sets, functions, fields, and complex numbers, respectively, are intended to review basic ideas used throughout the book. Appendix E on polynomials is used primarily in Chapters 5 and 7, especially in Section 7. We prefer to cite particular results from the appendices as needed rather than to discuss the appendices Preface xi independently. The following diagram illustrates the dependencies among the various chapters. Chapter 1? Chapter 2? Chapter 3? Sections 4.

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You can read this full article here: friedberg linear algebra pdf. Friedberg Linear Algebra by Louis Antoine Wolf, Gregory Kontorovich, David Eisenbud is a book written for graduate Illustrates the power of linear algebra through practical applications. This acclaimed theorem-proof text presents a careful treatment of the principal topics of linear algebra. It emphasizes A Visual Guide to the Friedberg Linear Algebra pdf download with a table of contents: 1. Diagonal matrices 2. Different types of division 3. Elementary row operations 4. Increasing and Linear Algebra Friedberg 5Th Edition PDF Download. The book, Linear Algebra Friedberg 5Th Edition, has been published. The information in this book is intended for those who are Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence-Linear Algebra, 4th Edition-Prentice Hall ().djvu Linear Algebra, 4th edition Linear Algebra This page intentionally left blank Fourth Edition Stephen H. Friedberg Arnold J. Insel Lawrence E. S. 26, 3, 3MB. Pages Page size ... read more

Using e of Theorem 2. Canonical forms are treated in Chapter 7. These points determine a unique plane, and its equation can be found by use of our previous observations about vectors. It is easily checked that T is an isomorphism; so F2 is isomorphic to P1 F. The main reason for this is that most of the important geometrical transformations are linear. If we have a relationship on a set of n objects that we denote by 1, 2,. How many suites were sold during the June sale?

Nevertheless, we repeat some of the definitions for use in this section. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. CONTACT Schamberger Freeway Apt. By use of the Lagrange interpolation formula in Section 1. There is, however, a natural way to combine two subspaces W1 and W2 to obtain a subspace that contains both W1 and W2, linear algebra friedberg pdf free download. Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another.