Introduction to linear algebra 4th edition pdf download

The dart that appears on the cover of this book symbolizes a vector and reflects my conviction that geometric understanding should precede computational techniques. I have tried to limit the number of theorems in the text. For the most part, results labeled as theorems either will be used later in the text or summarize preceding work. Interesting results that are not central to the book have been included as exercises or explorations. For example, the cross product of vectors is discussed only in explorations in Chapters 1 and 4.

Unlike most linear algebra textbooks, this book has no chapter on determinants. The essential results are all in Section 4. In the second edition, a new chapter on Jordan normal form was added which reappears here in expanded form as the second goal of this new edition, after the principal axis theorem.

To achieve these goals in one semester it is necessary to follow a straight path, but this is compensated by a wide selection of examples and exercises. In addition, the author includes an introduction to invariant theory to show that linear algebra alone is incapable of solving these canonical forms problems.

A compact, but mathematically clean introduction to linear algebra with particular emphasis on topics in abstract algebra, the theory of differential equations, and group representation theory. This new book offers a fresh approach to matrix and linear algebra by providing a balanced blend of applications, theory, and computation, while highlighting their interdependence.

Intended for a one-semester course, Applied Linear Algebra and Matrix Analysis places special emphasis on linear algebra as an experimental science, with numerous examples, computer exercises, and projects. While the flavor is heavily computational and experimental, the text is independent of specific hardware or software platforms. Throughout the book, significant motivating examples are woven into the text, and each section ends with a set of exercises.

A subalgebra, A1, of an algebra A is a linear subspace which is closed under the multiplication in A; that is, if x and y are arbitrary elements of A1, then xyeA1. Thus A inherits the structure of an algebra from A. It is clear that a subalgebra of an associative commutative algebra is itself associative commutative. Let S be a subset of A, and suppose that A is associative. Sr, clearly a subalgebra of A, called the subalgebra generated by S. Chapter V. A subspace that is both a right and left ideal is called a tivt'o-sided ideal, or simply an ideal in A.

Clearly, every right left ideal is a subalgebra. As an example of an ideal, consider the subspace A2 linearly generated by the products xy.

A2 is clearly an ideal and is called the derived algebra. The ideal I generated by a set S is the intersection of all ideals containing S. In particular every single clement a generates an ideal principal ideal generated by a. It follows that p is a monomorphism. Consequently we may identify F with its image under p. Then F becomes a subalgebra of A and scalar multiplication coincides with algebra multiplication.

In fact, if. This shows that Na is a right ideal in A. It is called the right annihilator of a. Similarly the left annihilator of a is defined. Factor algebras. Let A be an algebra and B be an arbitrary subspace of A. Conversely, assume B is an ideal. It has to be shown that the above product does not depend on the choice of x and y.

Then x'—xeB and y'—yeB. Finally, rewriting 5. Suppose A and B are algebras and is a homomorphism. Then the kernel of p is an ideal in A.

In the same way it follows that yxEker tp. Next consider the subspace Im pcB. Algebras Now let be the induced injective linear mapping. Finally, assume that C is a third algebra, and let be a homomorphism. In fact, we have 'ps.

E S, v is an arbitrary element we have that Basic properties be an arbitrary set map. Conversely, assume that 5. It follows from 5.

Algebras 5. Then the elementary rules of calculus imply that 0 is a derivation. If A has a unit element e it follows from 5. A derivation is completely determined by its action on a system of generators of A, as follows from an argument similar to that used to prove the same result for homomorphisms. For every derivation 0 we have the Leibniz formula where on xy 5. Suppose now by induction that 5.

Basic properties The image of a derivation 0 in A is of course a subspace of A, but it is in general not a subalgebra. Similarly, the kernel is a subalgebra, but it is not, in general, an ideal.

Let A and B be algebras and 'p : A B be a fixed homomorphism. Recall that an involution in a linear space is a linear transformation whose square is the identity. Similarly we define an involution w in an algebra A to be an endomorphisin of A whose square is the identity map. Clearly the identity map of A is an involution. If A has a unit element e it follows from sec.

Now let w be a fixed involution in A. As in the case of a derivation it is easy to show that an antiderivation is determined by its action on a system of generators for A and that ker Q is a subalgebra of A. It also follows easily that any linear combination of antiderivations with respect to a fixed involution w is again an antiderivation with respect to w. Suppose next that Q1 and Q2 are antiderivations in A with respect to 0w1. Then w1 oW2 the involutions w1 and w2 and assume that w1 0w2 is again an involution.

Then the relation shows that with respect to the involution W1 w2. Assume that WA is an involution in A. Show that C A is a subspace of A. If A is associative, prove that C A is a subalgebra of A. C A is called the centre of A. If A is any algebra and 0 is a derivation in A, prove that C A and the derived algebra are stable under 0. Construct an explicit example to prove that the sum of two endomorphisms is in general not an endomorphism.

Prove that 2p is a homomorphism if and only if the derived algebra is contained in ker 'p. Let C1 and C denote respectively the algebras of continuously differentiable and continuous functionsf: cf. Example 4. Suppose A is an associative commutative algebra and 0 is a derivation in A. Suppose that 0 is a derivation in an associative commutative algebra A with identity e and assume that xeA is invertible; i.

Let L be an algebra in which the product of two elements x, y is denoted by [x, y]. Let Ad a be the multiplication operator in the Lie algebra L. Prove that Ad a is a derivation. Let A be an associative algebra with product xy. Let A be any algebra and consider the space D A of derivations in A.

Prove that D A is a Lie algebra. Determine the kernel of p. Basic properties Let E be a finite dimensional vector space. Show that p is a monomorphism. Conversely, given n2 linear transformations of E satisfying i and ii , prove that they form a basis of L E; E and are induced by a basis of E. Define an equivalence relation in the set of all linearly independent E , in the following way: n2-tuples p Prove that the bases of L E; E defined in problem 14 form an equivalence class under the equivalence relation of problem Let A be an associative algebra, and let L denote the corresponding Chapter V.

Algebras is Lie algebra cf. Show that a linear mapping derivation in A only if it is a derivation in L. Conversely, prove that every derivation in A E; E is of this form. Use problem Ideals 5.

The lattice of ideals. Let A be an algebra, and consider the set 1 of ideals in A. We order this set by inclusion; i. Now let and '2 be ideals in A. Nilpotent ideals. Let A be an associative algebra. An ideal I will be called nilpotent if for some k, ,k0 5. Let A be an associative commutative algebra. Then the nilpotent elements of A form an ideal. Ideals The ideal consisting of the nilpotent elements is called the radical of A and will be denoted by rad A.

The definition of radical can be generalized to the non-commutative case; the theory is then much more difficult and belongs to the theory of rings and algebras. The reader is referred to [14]. To prove this assume that A is an element such that for some k.

Now assume that the algebra A has dimension n. Then rad A is a nilpotent ideal, and 5. For the proof, we choose a basis e1, Then each is nilpotent. Prove that every normal non-selfadjoint transformation of a plane is homothetic. Prove that is then linear. Let be a linear automorphism of an n-dimensional real linear space E. Show that an inner product can be defined in E such that p becomes a rotation if and only if the following conditions are fulfilled: i The space E can be decomposed into stable planes and stable straight lines.

Consider a proper rotation t which commutes with all proper rotations. Proper rotations of the plane. Let E be an oriented Euclidean plane and let zl denote the normed positive determinant function in E.

Chapter VIII. In fact, 1 follows directly from the definition. To verify 2 observe that the identity 7. Next, let p be any proper rotation of E. Fix a non-zero vector x and denote by 0 the oriented angle between x and p x cf. Inserting 8. The second relation is proved in a similar way. Rotations of Euclidean spaces of dimension 2, 3 and 4 Relations 8. It is called the rotation angle of and is denoted by e p.

Now 8. Remark: If E is a non-oriented plane we can still assign a rotation angle to a proper rotation p. Proper rotations of 3-space. Consider a proper rotation p of a 3-dimensional inner product space E. As has been shown in sec. In fact, assume that a and b are two linearly independent invariant vectors.

Then the invariant vectors generate a i-dimensional subspace E1 called the axis of p. Linear mappings of inner product spaces To determine the axis of a given rotation p consider the skew mapping 8. The vector u which is uniquely determined by p iS called the rotation-vector. The rotation vector is contained in the axis of p.

Then equations 8. Hence 8. The rotation angle. Consider the plane F which is orthogonal to E1. Then transforms F into itself and the induced rotation is again proper. Denote by 0 the rotation angle of p1 cf. Then, in view of 8. To find a formula for sin 0 consider the orientation of Fwhicl' is induced by E and by the vector u cf. Then formula 7. Rotations of Euclidean spaces of dimension 2.

This equation shows that sin 0 is positive and hence that 0 1 and define polynomials by — g are relatively prime, and hence there exist polynomials h1 such that Theory of a linear transformation be the decomposition of x determined by Then Arbitrary stable subspaces. Let Fc: E be any stable subspace. In fact, since the projection operators are polynomials in go, it follows that F is stable under each 7C1.

The Fitting decomposition. Let F1 be the direct sum of the remaining generalized eigenspaces. F0 and F1 are called respectively the Fitting-null component and the Fitting-one component of E. Generalized eigenspaces Clearly F0 and F1 are stable subspaces. Finally, we remark that the corresponding projection operators are polynomials in p, since they are sums of the projection operators defined in sec.

Then if f is any polynomial, it follows from sec. In particular, the minimum polynomials of 'p and coincide. Now suppose that F is any stable subspace of E. This proves that F' is stable. Generalized eigenspaces are stable under Then, as shown above, the and Now a comparison of the decompositions Show that the minimum polynomial of p is completely reducible i.

Use problem 3 to derive a simple proof of the basis deformation theorem of sec. For each transformation a Construct the decomposition of into the generalized eigenspaces. Use this result to show that and are dual and to obtain formula where the Let K1, denote the kernel of ar,. K,, is an ideal in T[t]. Clearly the minimum polynomial of p. Chapter XIII.

Theory of a linear transformation Then there is a vector tIE E such that Suppose now that hEKa and let g be the greatest common divisor of h and p. Now it follows from Let is the induced transformation. Then the minimum polynomial of given by cf. Now set 1 l1 Assume that jEK Cyclic spaces. Every such vector is called a geiiclaloI of E. Then, Ifa is a generator of E. Proof Let a be a generator of E.

It follows that cf. Proposition I, sec. Then the vectors a,p a form a basis of E. Then these vectors generate E. On the other hand. Cyclic subspaces. Every vector UEE determines a cyclic subspace. Throic;,, I: The degree of the minimum polynomial satisfies degp dimE. Equality holds if and only if E is cyclic. Moreover, if F is any cyclic subspace of E. Proof: Proposition V implies that deg p dimE. If E is cyclic. Proposition III. Cyclic spaces and irreducible spaces Finally, let Fc:E be any cyclic subspace and let v denote the minimum polynomial of the induced transformation.

Corollary: Let F E be any cyclic subspace, and let v denote the minimum polynomial of the linear transformation induced in F by p. Decomposition of E into cyclic subspaces. Theorem II: There exists a decomposition of E into a direct sum of cyclic subspaces. Proof: The theorem is an immediate consequence with the aid of an induction argument on the dimension of E of the following lemma. E such that Lemma I. Then cf. Theory of a linear transformations is dual to with respect to the induced scalar product between Ea and The corollary to Theorem I, sec.

Hence cf. Thus Theorem I, sec. Now consider the cyclic subspace Since I it follows that On the other hand. Theorem I. Irreducible spaces A vector space E is called! Introduction to Linear Algebra, Fourth Edition includes challenge problems to complement the review problems that have been highly praised in previous editions. The basic course is followed by seven applications: differential equations, engineering, graph theory, statistics, Fourier methods and the FFT, linear programming, and computer graphics.

There are no reviews yet. Be the first one to write a review. Structure of the Textbook Already in this preface, you can see the style of the book and its goal. That goal is serious, to explain this beautiful and useful part of mathematics. You will see how the applications of linear algebra reinforce the key ideas.

I hope every teacher will learn something new; familiar ideas can be seen in a new way. The book moves gradually and steadily from numbers to vectors to subspaces—each level comes naturally and everyone can get it.

Here are ten points about the organization of this book: 1 Chapter 1 starts with vectors and dot products. If the class has met them before, focus quickly on linear combinations. The new Section 1. Those two examples are the beginning of linear algebra. The heart of linear algebra is in that connection between the rows of A and the columns: the same numbers but very different pictures.

Then begins the algebra of matrices: an elimination matrix E multiplies A to produce a zero. The goal here is to capture the whole process—start with A and end with an upper triangular U.

The lower triangular L holds all the forward elimination steps, and U is the matrix for back substitution. Chapter 3 is linear algebra at the best level: subspaces. The column space contains all linear combinations of the columns. The crucial question is: How many of those columns are needed? The answer tells us the dimension of the column space, and the key information about A.

We reach the Fundamental Theorem of Linear Algebra. Chapter 4 has m equations and only n unknowns. We cannot throw out equations that are close but not perfectly exact. When we solve by least squares, the key will be the matrix A1 A. This wonderful matrix AJA appears everywhere in applied mathematics, when A is rectangular.

Determinants in Chapter 5 give formulas for all that has come before—inverses, pivots, volumes in n-dimensional space, and more. We don't need those formulas to compute! They slow us down. VIII Preface 6. Section 6. Many courses want to see eigenvalues early. It is completely reasonable to come here directly from Chapter 3, because the determinant is easy for a 2 by 2 matrix.

In those special directions A acts like a single number the eigenvalue A and the problem is one-dimensional. Chapter 6 is full of applications. One highlight is diagonalizing a symmetric matrix. Another highlight—not so well known but more important every day—is the diagonalization of any matrix.

This needs two sets of eigenvectors, not one, and they come of course! This Singular Value Decomposition often marks the end of the basic course and the start of a second course. Chapter 7 explains the linear transformation approach—it is linear algebra without coordinates, the ideas without computations.

Then Chapter 10 movesfrom real numbers and vectors to complex vectors and matrices. The Fourier matrix F is the most important complex matrix we will ever see.

Chapter 8 is full of applications, more than any single course could need: 8. Every section in the basic course ends with a Review of the Key Ideas. How should computing be included in a linear algebra course? It can open a new understanding of matrices—every class will find a balance. I chose the language of MATLAB as a direct way to describe linear algebra: eig ones 4 will produce the eigenvalues 4,0,0,0 of the 4 by 4 all-ones matrix.

Go to netlib. You can freely choose a different system. More and more software is open source. The new website math. Please contribute! Good problems are welcome by email: [email protected] Send new applications too, linear algebra is an incredibly useful subject. Preface The Variety of Linear Algebra Calculus is mostly about one special operation the derivative and its inverse the integral.

Of course I admit that calculus could be important But so many applications of mathematics are discrete rather than continuous, digital rather than analog. The century of data has begun! The truth is that vectors and matrices have become the language to know. Part of that language is the wonderful variety of matrices.