Linear algebra friedberg 4th edition pdf download

Linear algebra friedberg 4th edition pdf download

Linear Algebra – Stephen H. Friedberg – 4th Edition,Description

About the Linear Algebra 4th Edition Stephen H Friedberg Pdf Free Download. The linear algebra 4th edition pdf incorporates many features that have been requested by teachers over the 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 Fourth Edition Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Illinois State University PEARSON EDUCATION, Upper Saddle River, New Jersey Library of (PDF) Download Linear Algebra - Stephen H. Friedberg - 4th Edition You are at: Home Math Linear Algebra Linear Algebra – Stephen H. Friedberg – 4th Edition Linear Algebra – 11/10/ · DOWNLOAD LINEAR ALGEBRA FRIEDBERG 4TH ED linear algebra friedberg 4th pdf Until 19th century, linear algebra was introduced through systems of linear equations ... read more

Orthogonal Projections and the Spectral Theorem. The Singular Value Decomposition and the Pseudoinverse. Bilinear and Quadratic Forms. Einstein's Special Theory of Relativity. Conditioning and the Rayleigh Quotient. The Geometry of Orthogonal Operators. Canonical Forms. The Jordan Canonical Form I. The Jordan Canonical Form II. The Minimal Polynomial. Rational Canonical Form. Complex Numbers. Answers to Selected Exercises. Spence ISBN ISBN Edition: 4th Edition Topic: Math Subtopic: Linear Algebra File Type: eBook Solution Manual Idioma: English. Related subjects More of Linear Algebra. More books by: Stephen H.

Friedberg Arnold J. Insel Lawrence E. featured Friedberg Linear Algebra Linear Engineering Book Computer Solutions Solution Manuals Mechanics Textbooks Student problems Engineers Applications Power Solutions Manual Free. Download now Linear Algebra. Type of file. Linear Algebra: A Modern Introduction — David Poole — 2nd Edition. Linear Algebra and Its Applications — David C. Lay — 4th Edition. Elementary Linear Algebra — Ron Larson, David C. Falvo — 6th Edition. Lay — 3rd Edition. Linear Algebra and Its Applications — Gilbert Strang — 3rd Edition. Contemporary Linear Algebra — Howard Anton, Robert C. Busby — 1st Edition. Introduction to Linear Algebra — Gilbert Strang — 3rd Edition. Elementary Linear Algebra with Applications — Howard Anton, Chris Rorres — 9th Edition. Matrices Theory and Applications — Denis Serre — 1st Edition. Robert Jacobs, Richard Chase — 12th Edition. Leave us a comment No Comments. Connect with D.

I allow to create an account. When you login first time using a Social Login button, we collect your account public profile information shared by Social Login provider, based on your privacy settings. We also get your email address to automatically create an account for you in our website. Once your account is created, you'll be logged-in to this account. Disagree Agree. Review this book:. Inline Feedbacks. Let u and v denote vectors beginning at A and ending at B and C, respectively. The endpoint of su is the point of intersection of the line through A and B with the line through S Sec. A similar procedure locates the endpoint of tv. In the next section we formally define a vector space and consider many examples of vector spaces other than the ones mentioned above. EXERCISES 1. Determine whether the vectors emanating from the origin and terminating at the following pairs of points are parallel. Find the equations of the lines through the following pairs of points in space.

Find the equations of the planes containing the following points in space. What are the coordinates of the vector 0 in the Euclidean plane that satisfies property 3 on page 3? Justify your answer. Prove that if the vector x emanates from the origin of the Euclidean plane and terminates at the point with coordinates a1 , a2 , then the vector tx that emanates from the origin terminates at the point with coordinates ta1 , ta2. Prove that the diagonals of a parallelogram bisect each other. Many other familiar algebraic systems also permit definitions of addition and scalar multiplication that satisfy the same eight properties.

In this section, we introduce some of these systems, but first we formally define this type of algebraic structure. A vector space or linear space V over a field 2 F consists of a set on which two operations called addition and scalar multiplication, respectively are defined so that for each pair of elements x, y, 2 Fields are discussed in Appendix C. The elements of the field F are called scalars and the elements of the vector space V are called vectors. A vector space is frequently discussed in the text without explicitly mentioning its field of scalars. The reader is cautioned to remember, however, that every vector space is regarded as a vector space over a given field, which is denoted by F. Occasionally we restrict our attention to the fields of real and complex numbers, which are denoted R and C, respectively.

Observe that VS 2 permits us to unambiguously define the addition of any finite number of vectors without the use of parentheses. In the remainder of this section we introduce several important examples of vector spaces that are studied throughout this text. Observe that in describing a vector space, it is necessary to specify not only the vectors but also the operations of addition and scalar multiplication. An object of the form a1 , a2 ,. The elements 8 Chap. Two n-tuples a1 , a2 ,. Example 1 The set of all n-tuples with entries from a field F is denoted by Fn. Thus R3 is a vector space over R. Similarly, C2 is a vector space over C. Since a 1-tuple whose only entry is from F can be regarded as an element of F , we usually write F rather than F1 for the vector space of 1-tuples with entry from F. The entries ai1 , ai2 ,. The rows of the preceding matrix are regarded as vectors in Fn , and the columns are regarded as vectors in Fm.

The m × n matrix in which each entry equals zero is called the zero matrix and is denoted by O. In addition, if the number of rows and columns of a matrix are equal, the matrix is called square. Note that these are the familiar operations of addition and scalar multiplication for functions used in algebra and calculus. When F is a field containing infinitely many scalars, we usually regard a polynomial with coefficients from F as a function from F into F. See page With these operations of addition and scalar multiplication, the set of all polynomials with coefficients from F is a vector space, which we denote by P F. Example 5 Let F be any field. A sequence in F is a function σ from the positive integers into F. is denoted {an }. Let V consist of all sequences {an } in F that have only a finite number of nonzero terms an. With these operations V is a vector space. Since VS 1 , VS 2 , and VS 8 fail to hold, S is not a vector space with these operations. Then S is not a vector space with these operations because VS 3 hence VS 4 and VS 5 fail.

Theorem 1. Corollary 1. The vector 0 described in VS 3 is unique. Corollary 2. The vector y described in VS 4 is unique. The next result contains some of the elementary properties of scalar multiplication. The proof of c is similar to the proof of a. Label the following statements as true or false. a Every vector space contains a zero vector. b A vector space may have more than one zero vector. e A vector in Fn may be regarded as a matrix in Mn×1 F. f An m × n matrix has m columns and n rows. g In P F , only polynomials of the same degree may be added. i If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n. k Two functions in F S, F are equal if and only if they have the same value at each element of S.

Write the zero vector of M3×4 F. Perform the indicated operations. Wildlife Management, 25, reports the following number of trout having crossed beaver dams in Sagehen Creek. Upstream Crossings Brook trout Rainbow trout Brown trout Fall Spring Summer 8 3 3 3 0 0 1 0 0 14 Chap. At the end of May, a furniture store had the following inventory. Early Mediter- American Spanish ranean Danish Living room suites Bedroom suites 4 5 2 1 1 1 3 4 Dining room suites 3 1 2 6 Record these data as a 3 × 4 matrix M. To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2M. How many suites were sold during the June sale?

Prove Corollaries 1 and 2 of Theorem 1. Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3. Prove that V is a vector space over F. V is called the zero vector space. Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space. Let V denote the set of ordered pairs of real numbers. Is V a vector space over R with these operations?

n}; so V is a vector space over C by Example 1. Is V a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication? n}; so V is a vector space over R by Example 1. Is V a vector space over the field of complex numbers with the operations of coordinatewise addition and multiplication? Let V denote the set of all m × n matrices with real entries; so V is a vector space over R by Example 2. Let F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix addition and scalar multiplication?

Is V a vector space over F with these operations? c Is V a vector space over R with these operations? Let V be the set of sequences {an } of real numbers. See Example 5 for the definition of a sequence. Prove that, with these operations, V is a vector space over R. Let V and W be vector spaces over a field F. How many matrices are there in the vector space Mm×n Z2? See Appendix C. The appropriate notion of substructure for vector spaces is introduced in this section. A subset W of a vector space V over a field F is called a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication defined on V.

In any vector space V, note that V and {0 } are subspaces. The latter is called the zero subspace of V. Fortunately it is not necessary to verify all of the vector space properties to prove that a subset is a subspace. Because properties VS 1 , VS 2 , VS 5 , VS 6 , VS 7 , and VS 8 hold for all vectors in the vector space, these properties automatically hold for the vectors in any subset. Thus a subset W of a vector space V is a subspace of V if and only if the following four properties hold. W is closed under addition. W is closed under scalar multiplication. W has a zero vector. Each vector in W has an additive inverse in W. The next theorem shows that the zero vector of W must be the same as the zero vector of V and that property 4 is redundant. Let V be a vector space and W a subset of V.

Then W is a subspace of V if and only if the following three conditions hold for the operations defined in V. If W is a subspace of V, then W is a vector space with the operations of addition and scalar multiplication defined on V. So condition a holds. Conversely, if conditions a , b , and c hold, the discussion preceding this theorem shows that W is a subspace of V if the additive inverse of each vector in W lies in W. Hence W is a subspace of V. The preceding theorem provides a simple method for determining whether or not a given subset of a vector space is a subspace. Normally, it is this result that is used to prove that a subset is, in fact, a subspace. For example, the 2 × 2 matrix displayed above is a symmetric matrix. Clearly, a symmetric matrix must be square. The set W of all symmetric matrices in Mn×n F is a subspace of Mn×n F since the conditions of Theorem 1.

The zero matrix is equal to its transpose and hence belongs to W. See Exercise 3. Using this fact, we show that the set of symmetric matrices is closed under addition and scalar multiplication. The examples that follow provide further illustrations of the concept of a subspace. The first three are particularly important. Example 1 Let n be a nonnegative integer, and let Pn F consist of all polynomials in P F having degree less than or equal to n. Moreover, the sum of two polynomials with degrees less than or equal to n is another polynomial of degree less than or equal to n, and the product of a scalar and a polynomial of degree less than or equal to n is a polynomial of degree less than or equal to n.

So Pn F is closed under addition and scalar multiplication. It therefore follows from Theorem 1. Clearly C R is a subset of the vector space F R, R defined in Example 3 of Section 1. We claim that C R is a subspace of F R, R. Moreover, the sum of two continuous functions is continuous, and the product of a real number and a continuous function is continuous. So C R is closed under addition and scalar multiplication and hence is a subspace of F R, R by Theorem 1. Clearly the zero matrix is a diagonal matrix because all of its entries are 0. Therefore the set of diagonal matrices is a subspace of Mn×n F by Theorem 1. Any intersection of subspaces of a vector space V is a subspace of V. Let C be a collection of subspaces of V, and let W denote the intersection of the subspaces in C. Then x and y are contained in each subspace in C. Having shown that the intersection of subspaces of a vector space V is a subspace of V, it is natural to consider whether or not the union of subspaces of V is a subspace of V.

It is easily seen that the union of subspaces must contain the zero vector and be closed under scalar multiplication, but in general the union of subspaces of V need not be closed under addition. In fact, it can be readily shown that the union of two subspaces of V is a subspace of V if and only if one of the subspaces contains the other. See Exercise There is, however, a natural way to combine two subspaces W1 and W2 to obtain a subspace that contains both W1 and W2. As we already have suggested, the key to finding such a subspace is to assure that it must be closed under addition. This idea is explored in Exercise a If V is a vector space and W is a subset of V that is a vector space, then W is a subspace of V. b The empty set is a subspace of every vector space. d The intersection of any two subsets of V is a subspace of V. f The trace of a square matrix is the product of its diagonal entries. Determine the transpose of each of the matrices that follow.

In addition, if the matrix is square, compute its trace. Prove that diagonal matrices are symmetric matrices. Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3. Justify your answers. Let W1 , W3 , and W4 be as in Exercise 8. Prove that the upper triangular matrices form a subspace of Mm×n F. Let S be a nonempty set and F a field. Prove that C S, F is a subspace of F S, F. Is the set of all differentiable real-valued functions defined on R a subspace of C R?

Let Cn R denote the set of all real-valued functions defined on the real line that have a continuous nth derivative. Prove that Cn R is a subspace of F R, R. Let W1 and W2 be subspaces of a vector space V. Show that the set of convergent sequences {an } i. Let F1 and F2 be fields. Prove that the set of all even functions in F F1 , F2 and the set of all odd functions in F F1 , F2 are subspaces of F F1 , F2. W1 is the set of all upper triangular matrices defined in Exercise Let V denote the vector space consisting of all upper triangular n × n matrices as defined in Exercise 12 , and let W1 denote the subspace of V consisting of all diagonal matrices. Clearly, a skewsymmetric matrix is square. Let F be a field. Prove that the set W1 of all skew-symmetric n × n matrices with entries from F is a subspace of Mn×n F.

Now assume that F is not of characteristic 2 see Appendix C , and let W2 be the subspace of Mn×n F consisting of all symmetric n × n matrices. Let F be a field that is not of characteristic 2. Both W1 and W2 are subspaces of Mn×n F. Compare this exercise with Exercise Let W be a subspace of a vector space V over a field F. d Prove that the set S is a vector space with the operations defined in c. An important special case occurs when A is the origin. This is proved as Theorem 1. The appropriate generalization of such expressions is presented in the following definitions. Let V be a vector space and S a nonempty subset of V. In this case we also say that v is a linear combination of u1 , u2 ,.

Thus the zero vector is a linear combination of any nonempty subset of V. Example 1 TABLE 1. Watt and Annabel L. Merrill, Composition of Foods Agriculture Handbook Number 8 , Consumer and Food Economics Research Division, U. Department of Agriculture, Washington, D. a Zeros in parentheses indicate that the amount of a vitamin present is either none or too small to measure. So grams of cupcake, grams of coconut custard pie, grams of raw brown rice, and grams of soy sauce provide exactly the same amounts of the five vitamins as grams of raw wild rice.

This question often reduces to the problem of solving a system of linear equations. In Chapter 3, we discuss a general method for using matrices to solve any system of linear equations. To solve system 1 , we replace it by another system with the same solutions, but which is easier to solve. The procedure to be used expresses some of the unknowns in terms of others by eliminating certain unknowns from all the equations except one. This need not happen in general. We now want to make the coefficient of a3 in the second equation equal to 1, and then eliminate a3 from the third equation. The procedure just illustrated uses three types of operations to simplify the original system: 1. interchanging the order of any two equations in the system; 2. multiplying any equation in the system by a nonzero constant; 3.

adding a constant multiple of any equation to another equation in the system. In Section 3. Note that we employed these operations to obtain a system of equations that had the following properties: 1. The first nonzero coefficient in each equation is one. If an unknown is the first unknown with a nonzero coefficient in some equation, then that unknown occurs with a zero coefficient in each of the other equations. The first unknown with a nonzero coefficient in any equation has a larger subscript than the first unknown with a nonzero coefficient in any preceding equation. Once a system with properties 1, 2, and 3 has been obtained, it is easy to solve for some of the unknowns in terms of the others as in the preceding example.

See Example 2. We return to the study of systems of linear equations in Chapter 3. We discuss there the theoretical basis for this method of solving systems of linear equations and further simplify the procedure by use of matrices. We now name such a set of linear combinations. Let S be a nonempty subset of a vector space V. The span of S, denoted span S , is the set consisting of all linear combinations of the vectors in S. Thus the span of { 1, 0, 0 , 0, 1, 0 } contains all the points in the xy-plane. In this case, the span of the set is a subspace of R3. This fact is true in general. The span of any subset S of a vector space V is a subspace of V.

Moreover, any subspace of V that contains S must also contain the span of S. Then there exist vectors u1 , u2 ,. Thus span S is a subspace of V. Now let W denote any subspace of V that contains S. In this case, we also say that the vectors of S generate or span V. So any linear combination of these matrices has equal diagonal entries. Hence not every 2 × 2 matrix is a linear combination of these three matrices. It is natural to seek a subset of W that generates W and is as small as possible. In the next section we explore the circumstances under which a vector can be removed from a generating set to obtain a smaller generating set. a The zero vector is a linear combination of any nonempty set of vectors. c If S is a subset of a vector space V, then span S equals the intersection of all subspaces of V that contain S. d In solving a system of linear equations, it is permissible to multiply an equation by any constant. e In solving a system of linear equations, it is permissible to add any multiple of one equation to another.

f Every system of linear equations has a solution. Solve the following systems of linear equations by the method introduced in this section. For each of the following lists of vectors in R3 , determine whether the first vector can be expressed as a linear combination of the other two. For each list of polynomials in P3 R , determine whether the first polynomial can be expressed as a linear combination of the other two. In each part, determine whether the given vector is in the span of S. Show that the vectors 1, 1, 0 , 1, 0, 1 , and 0, 1, 1 generate F3. In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {e1 , e2 ,. Show that Pn F is generated by {1, x,. Interpret this result geometrically in R3.

The sum of two subsets is defined in the exercises of Section 1. Let S1 and S2 be subsets of a vector space V. Let V be a vector space and S a subset of V with the property that whenever v1 , v2 ,. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S. Let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S generates W? Unless W is the zero subspace, W is an infinite set. Indeed, the smaller that S is, the fewer computations that are required to represent vectors in W. Let us attempt to find a proper subset of S that also generates W. The search for this subset is related to the question of whether or not some vector in S is a linear combination of the other vectors in S.

The reader should verify that no such solution exists. This does not, however, answer our question of whether some vector in S is a linear combination of the other vectors in S. By formulating our question differently, we can save ourselves some work. That is, because some vector in S is a linear combination of the others, the zero vector can be expressed as a linear combination of the vectors in S using coefficients that are not all zero. The converse of this statement is also true: If the zero vector can be written as a linear combination of the vectors in S in which not all the coefficients are zero, then some vector in S is a linear combination of the others.

Thus, rather than asking whether some vector in S is a linear combination of the other vectors in S, it is more efficient to ask whether the zero vector can be expressed as a linear combination of the vectors in S with coefficients that are not all zero. This observation leads us to the following definition. A subset S of a vector space V is called linearly dependent if there exist a finite number of distinct vectors u1 , u2 ,. In this case we also say that the vectors of S are linearly dependent. For any vectors u1 , u2 ,. We call this the trivial representation of 0 as a linear combination of u1 , u2 ,.

Thus, for a set to be linearly dependent, there must exist a nontrivial representation of 0 as a linear combination of vectors in the set. We show that S is linearly dependent and then express one of the vectors in S as a linear combination of the other vectors in S. To show that Sec. 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.

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Linear Algebra, 4th Edition, introduces linear algebra in a logical, well-motivated sequence. Its careful explanations of core ideas are followed by rigorous development of advanced topics. The linear algebra 4th edition pdf incorporates many features that have been requested by teachers over the years: there is a large amount of material on linear algebra used as a basis for understanding differential equations, as well as a clearer treatment of topics such as determinants, including sections on adjoints and minors. The exercises have been written with the needs of the instructor in mind — many are new — and solutions manuals are available at a reasonable cost.

linear algebra 4th edition pdf is one of the best educational books that you will ever come across. It is a book that can serve both as a reference textbook and as course material for learning. For courses in Advanced Linear Algebra. 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. linear algebra 4th edition pdf — In 21 chapters, with exercises ranging from routine to some challenging problems. Linear algebra describes the basic operations of vector spaces and linear maps between them. The book presents the subject at a level that befits self-contained course material for the first-year sequence in both pure and applied mathematics or engineering. Vector Spaces 1. Linear Transformations and Matrices 2.

Elementary Matrix Operations and Systems of Linear Equations 3. Determinants 4. Diagonalization 5. Inner Product Spaces 6. Canonical Forms 7. Sets B. Functions C. Fields D. Complex Numbers E. Polynomials Answers to Selected Exercises Index. Stephen H. Friedberg holds a BA in mathematics from Boston University and MS and PhD degrees in mathematics from Northwestern University, and was awarded a Moore Postdoctoral Instructorship at MIT. He was a faculty member at Illinois State University for 32 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in He has also taught at the University of London, the University of Missouri, and at Illinois Wesleyan University. He has authored or coauthored articles and books in analysis and linear algebra. Arnold J. Insel received BA and MA degrees in mathematics from the University of Florida and a PhD from the University of California at Berkeley.

He served as a faculty member at Illinois State University for 31 years and at Illinois Wesleyan University for two years. In addition to authoring and co-authoring articles and books in linear algebra, he has written articles in lattice theory, topology, and topological groups. Lawrence E. Spence holds a BA from Towson State College and MS and PhD degrees in mathematics from Michigan State University. He served as a faculty member at Illinois State University for 34 years, where he was recognized as the outstanding teacher in the College of Arts and Sciences in He is an author or co-author of nine college mathematics textbooks, as well as articles in mathematics journals in the areas of discrete mathematics and linear algebra. Save my name, email, and website in this browser for the next time I comment.

Leave this field empty. com is dedicated to providing trusted educational content for students and anyone who wish to study or learn something new. It is a comprehensive directory of online programs, and MOOC Programs. Terms of Use. Privacy policy. Linear Algebra 4th Edition Stephen H Friedberg Pdf Free Download. I hope you find my information helpful. Have a wonderful reading. About the Linear Algebra 4th Edition Stephen H Friedberg Pdf Free Download The linear algebra 4th edition pdf incorporates many features that have been requested by teachers over the years: there is a large amount of material on linear algebra used as a basis for understanding differential equations, as well as a clearer treatment of topics such as determinants, including sections on adjoints and minors.

Polynomials Answers to Selected Exercises Index About the Author Stephen H. About the author. The Editorial Team at Infolearners. com is dedicated to providing the best information on learning. From attaining a certificate in marketing to earning an MBA, we have all you need. If you feel lost, reach out to an admission officer. Leave a Comment Cancel reply Comment Name Email Save my name, email, and website in this browser for the next time I comment. About us InfoLearners. Recommended Posts.

Friedberg, Insel, and Spence Linear algebra, 4th ed. SOLUTIONS REFERENCE,Related subjects

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 Fourth Edition Stephen H. Friedberg Arnold J. Insel Lawrence E. Spence Illinois State University PEARSON EDUCATION, Upper Saddle River, New Jersey Library of 11/10/ · DOWNLOAD LINEAR ALGEBRA FRIEDBERG 4TH ED linear algebra friedberg 4th pdf Until 19th century, linear algebra was introduced through systems of linear equations 3/06/ · DOWNLOAD LINEAR ALGEBRA 4TH EDITION BY FRIEDBERG SOLUTIONS linear algebra 4th edition pdf Buy Linear Algebra and Its Applications, 4th Edition on blogger.com About the Linear Algebra 4th Edition Stephen H Friedberg Pdf Free Download. The linear algebra 4th edition pdf incorporates many features that have been requested by teachers over the Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence-Linear Algebra, 4th Edition-Prentice Hall ().djvu - Google Drive ... read more

ALGA Terry Tao. c Prove that U is one-to-one, but not onto. Interpret this result geometrically in R3. Recall by Exercise 38 of Section 2. First, it defines the concept of a chain and a maximal set with respect to set inclusion. Using this fact, we show that the set of symmetric matrices is closed under addition and scalar multiplication.

Then compute the nullity and rank of T, linear algebra friedberg 4th edition pdf download, and verify the dimension theorem. Have a wonderful reading. Then there exist scalars a1. 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 Fmrespectively, then we have the following properties. Friedberg, Arnold J. Assume that S is a linearly dependent subset of a vector space V over a field F.