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Authors view affiliations Rami Shakarchi. Front Matter Pages i-xi. Vector Spaces. Pages Linear Mappings.
Linear Maps and Matrices. Back Matter Pages About this book Introduction Linear Algebra is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorems for linear maps, including eigenvectors and eigenvalues, quadric and hermitian forms, diagonalization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem.
However, the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added. Eigenvalue Eigenvector algebra linear algebra matrices matrix theory. Authors and affiliations Serge Lang 1 1. Reviews "The present textbook is intended for a one-term course at the junior or senior level. Buy options. This mixes both the computational and theoretical aspects. Determinants are treated much more briefly than in the first edition, and several proofs are omitted.
Students interested in theory can refer to a more complete treatment in theoretical books on linear algebra. I have included a chapter on eigenvalues and eigenvectors. This gives practice for notions studied previously, and leads into material which is used constantly in all parts of mathematics and its applications. I am much indebted to Toby Orloff and Daniel Horn for their useful comments and corrections as they were teaching the course from a pre- liminary version of this book.
I thank Allen Altman and Gimli Khazad for lists of corrections. Definition of Points in Space. Located Vectors. Scalar Prod uct. The Norm of a Vector. Parametric Lines. Multiplication of Matrices.
Homogeneous Linear Equations and Elimination. Row Operations and Gauss Elimination. Linear Combinations. Convex Sets. Linear Independence. The Rank of a Matrix. Linear Mappings. The Kernel and Image of a Linear Map. The Rank and Linear Equations Again. The Matrix Associated with a Linear Map. Composition of Linear Maps. Scalar Products. Orthogonal Bases. Bilinear Maps and Matrices. Determinants of Order 2. The Rank of a Matrix and Subdeterminants.
Cramer's Rule. Inverse of a Matrix. Determinants as Area and Volume. Eigenvectors and Eigenvalues. The Characteristic Polynomial. Eigenvalues and Eigenvectors of Symmetric Matrices. Diagonalization of a Symmetric Linear Map.
Complex Numbers. It provides geometric motivation for everything that follows. Hence the properties of vectors, both algebraic and geometric, will be discussed in full. One significant feature of all the statements and proofs of this part is that they are neither easier nor harder to prove in 3-space than they are in 2-space. Definition of Points in Space We know that a number can be used to represent a point on a line, once a unit length is selected.
A pair of numbers i. We simply introduce one more axis. Figure 2 illustrates this. Thus we can say that a single number represents a point in I-space. A couple represents a point in 2-space. A triple represents a point in 3- space. Although we cannot draw a picture to go further, there is nothing to prevent us from considering a quadruple of numbers. A quintuple would be a point in 5-space, then would come a sextuple, septuple, octuple, We let ourselves be carried away and define a point in n-space to be an n-tuple of numbers if n is a posItIve integer.
We shall denote such an n-tuple by a capital letter X, and try to keep small letters for numbers and capital letters for points. We call the numbers Xl' For example, in 3-space, 2 is the first coordinate of the point 2,3, -4 , and -4 is its third coordinate.
We denote n-space by Rn. Thus the reader may visualize either of these two cases throughout the book. However, three comments must be made.
Example 1. One classical example of 3-space is of course the space we live in. After we have selected an origin and a coordinate system, we can describe the position of a point body, particle, etc.
Furthermore, as was known long ago, it is convenient to extend this space to a 4-dimensional space, with the fourth coordinate as time, the time origin being selected, say, as the birth of Christ-although this is purely arbitrary it might be more convenient to select the birth of the solar system, or the birth of the earth as the origin, if we could deter- mine these accurately.
Then a point with negative time coordinate is a BC point, and a point with positive time coordinate is an AD point. Don't get the idea that "time is the fourth dimension ", however. The above 4-dimensional space is only one possible example. In economics, for instance, one uses a very different space, taking for coordinates, say, the number of dollars expended in an industry. For instance, we could deal with a 7-dimensional space with coordinates corresponding to the following industries: 1.
Steel 2. Auto 3. Farm products 4. Fish 5. Chemicals 6. Clothing 7. We agree that a megabuck per year is the unit of measurement. Then a point 1,, , , , , , in this 7-space would mean that the steel industry spent one billion dollars in the given year, and that the chemical industry spent mil- lion dollars in that year. The idea of regarding time as a fourth dimension is an old one. Already in the Encyclopedie of Diderot, dating back to the eighteenth century, d'Alembert writes in his article on "dimension": Cette maniere de considerer les quantites de plus de trois dimensions est aussi exacte que l'autre, car les lettres peuvent toujours etre regardees com me representant des nombres rationnels ou non.
J'ai dit plus haut qu'il n'etait pas possible de concevoir plus de trois dimensions. Un homme d'esprit de rna connaissance croit qu'on pourrait cependant regarder la duree comme une quatrieme dimension, et que Ie produit temps par la solidite serait en quelque maniere un produit de quatre dimensions; cette idee peut etre contestee, mais elle a, ce me semble, quelque merite, quand ce ne serait que celui de la nouveaute.
Encyclopedie, Vol. I said above that it was not possible to conceive more than three dimensions. A clever gentleman with whom I am acquainted believes that nevertheless, one could view duration as a fourth dimension, and that the product time by solidity would be somehow a product of four dimensions.
This idea may be chal- lenged, but it has, it seems to me, some merit, were it only that of being new. Observe how d'Alembert refers to a "clever gentleman" when he appar- ently means himeself. He is being rather careful in proposing what must have been at the time a far out idea, which became more prevalent in the twentieth century. D' Alembert also visualized clearly higher dimensional spaces as "prod- ucts" of lower dimensional spaces.
For instance, we can view 3-space as putting side by side the first two coordinates x l' x 2 and then the third x 3. Thus we write We use the product sign, which should not be confused with other "products", like the product of numbers.
The word "product" is used in two contexts. Similarly, we can write There are other ways of expressing R4 as a product, namely This means that we view separately the first two coordinates x l' x 2 and the last two coordinates X3' x 4.
We shall come back to such products later. We shall now define how to add points. All these properties are very simple, and are true because they are true for numbers, and addition of n-tuples is defined in terms of addition of their components, which are numbers. We usually denote this n-tuple by 0, and also call it zero, because no diffi- cuI ty can occur in practice. We shall now interpret addition and multiplication by numbers geo- metrically in the plane you can visualize simultaneously what happens in 3-space.
Example 3. We see again that the geometric representation of our addition looks like a parallelogram Fig. The above segments are therefore parallel and of the same length, as illustrated in Fig.
If we plot this point, we see that - A has opposite direction to A. We may view - A as the reflection of A through the origin. A -A Figure 6 We shall now consider multiplication of A by a number. If c is any number, we define cA to be the point whose coordinates are Example 6. I t is easy to verify the rules: 5. What is the geometric representation of multiplication by a number? Example 7. Multiplication by 3 amounts to stretching A by 3.
We emphasize that this means A and B have the same direction with respect to the origin. For simplicity of language, we omit the words "with respect to the origin". Mulitiplication by a negative number reverses the direction.
Thus - 3A would be represented as in Fig. Draw the points of Exercises 1 and 2 on a sheet of graph paper. Let A, B be as in Exercise 1.
Let A and B be as drawn in Fig. Draw the point A-B. Located Vectors We define a located vector to be an ordered pair of points which we This is not a product. We visualize this as an arrow be- tween A and B. We call A the beginning point and B the end point of the located vector Fig. Given any located vector AB , we shall say that it is located at A.
A located vector at the origin is entirely determined by its end point. In view of this, we shall call an n-tuple either a point or a vector, de- pending on the interpretation which we have in mind. In the next pictures, we illustrate parallel located vectors. In a similar manner, any definition made concerning n-tuples can be carried over to located vectors. For instance, in the next section, we shall define what it means for n-tuples to be perpendicular.
In Fig. Draw the located vectors of Exercises 1, 2, 5, and 6 on a sheet of paper to illustrate these exercises. Also draw the located vectors QP and BA. Scalar Product It is understood that throughout a discussion we select vectors always in the same n-dimensional space. For the moment, we do not give a geometric interpretation to this scalar product. We shall do this later. We derive first some important proper- ties. The basic ones are: SP t. If A, B, C are three vectors, then A.
We shall now prove these properties. This proves the first property. This proves what we wanted. We leave property SP 3 as an exercise. In much of the work which we shall do concerning vectors, we shall use only the ordinary properties of addition, multiplication by numbers, and the four properties of the scalar product. We shall give a formal discussion of these later. For the moment, observe that there are other objects with which you are familiar and which can be added, subtracted, and multiplied by numbers, for instance the continuous functions on an interval [a, bJ cf.
This is the only instance when we allow ourselves such a notation. Thus A 3 has no meaning. For the moment, it is not clear that in the plane, this definition coincides with our intuitive geometric notion of perpendicularity. We shall convince you that it does in the next section. Here we merely note an example. And these vectors look perpendicular. We see that A is perpendlcular to Ei according to our definition of perpendicularity with the dot product if and only if its i-th component is equal to 0.
Which of the following pairs of vectors are perpendicular? Let A be a vector perpendicular to every vector X. The norm IS also some- times called the magnitude of A. Example 2. Of course, this is as it should be from the picture: A -A Figure 17 Recall that A and - A are said to have opposite direction.
However, they have the same norm magnitude, as is sometimes said when speak- ing of vectors. Let A, B be two points. Thus our definitions reflect our geometric intuition derived from Pythagoras. These are illustrated in Fig. In higher dimensional space, one uses this same terminology of ball and sphere. Figure 21 illustrates a sphere and a ball in 3-space. The open ball consists of the region inside the shell excluding the shell itself. The closed ball consists of the region inside the shell and the shell itself.
We verify this formally using our definition of the length. Theorem 4. Taking the square root now yields what we want. Let S 1 be the sphere of radius 1, centered at the origin.
In this manner, we get all points of the sphere of radius a. Thus the sphere of radius a is obtained by stretching the sphere of radius 1, through multiplication by a. A similar remark applies to the open and closed balls of radius a, they being obtained from the open and closed balls of radius 1 through multiplication by a. If a -:f. Example 4. There are as many unit vectors as there are directions. General Pythagoras theorem. If A and B are perpendicular, then The theorem is illustrated on Fig.
We shall now use the notion of perpendicularity to derive the notion of projection. Hence we have seen that there is a unique number c such that A - cB is perpendicular to B, and c is given by the above formula. Th e projection fl' h A. Example 6. Our construction gives an immediate geometric interpretation for the scalar product. This is subject to the following disad- vantages, not to say objections: a The four properties of the scalar product SP 1 through SP 4 are then by no means obvious.
In higher dimensional space, it fails even more. Thus we prefer to lay obvious algebraic foundations, and then recover very simply all the properties. After working out some examples, we shall prove the inequality which allows us to justify this in n-space. First note a special case. We don't have to deal only with the special unit vector as above.
Let E be any unit vector, that is a vector of norm 1. Let c be the compon- ent of A along E. Let A, B be two vectors in Rn. In view of Theorem 4. The inequality of Theorem 4. Let A, B be vectors. In view of our previous result, this satisfies the inequality and the right-hand side is none other than Our theorem is proved. The reason for this is that if we draw a triangle as in Fig. All the proofs do not use coordinates, only properties SP 1 through SP 4 of the dot product.
Hence they remain valid in more gen- eral situations, see Chapter VI. In n-space, they give us inequalities which are by no means obvious when expressed in terms of coordinates.
For instance, the Schwarz inequality reads, in terms of coordinates: Just try to prove this directly, without the "geometric" intuition of Pyth- agoras, and see how far you get. Find the norm of the vector A in the following cases. Find the norm of vector B in the above cases. Find the projection of A along B in the above cases.
Find the projection of B along A in the above cases. Find the cosine between the following vectors A and B. Determine the cosine of the angles of the triangle whose vertices are a 2, -1,1 , 1, -3, -5 , 3, -4, Let AI' Interpret a as a "parallelogram law".
Let A, B, C be three non-zero vectors. At a given time t, the bug is at the point. This parametric representation is also useful to describe the set of points lying on the line segment between two given points. Let P, Q be two points. As t goes from 0 to 1, the bug goes from P to Q. Find the coordi- nates of the point which lies one third of the distance from P to Q.
Let S t as above be the parametric representation of the segment from P to Q. We first have to find a vector in the direction of the line. We shall now discuss the relation between a parametric representation and the ordinary equation of a line in the plane.
Suppose that we work in the plane, and write the coordinates of a point X as x, y. We can then eliminate t and obtain the usual equation relating x and y. We can always select at least one. Find a parametric representation for the line passing through the following pairs of points. Determine the coordinates of the fol- lowing points: a The midpoint of the line segment between P and Q.
If P, Q are two arbitrary points in n-space, give the general formula for the midpoint of the line segment between P and Q. Planes We can describe planes in 3-space by an equation analogous to the single equation of the line. We proceed as follows. We shall also say that this plane is the one perpendicular to N, and consists of all vectors X such that X - P is perpendicular to N. We have drawn a typical situation in 3-spaces in Fig.
Instead of saying that N is perpendicular to the plane, one also says that N is normal to the plane. Let t be a number o. Thus we may say that our plane is the plane passing through P and perpendicular to the line in the direction of N.
To find the equation of the plane, we could use any vector tN with t 0 instead of N. Then X. They give rise to a vector perpendicular to the line. If we want to find a point in that plane, we of course have many choices. We can give arbitrary val- ues to x and y, and then solve for z. Thus 1, 1,1 is a point in the plane. N is said to be the equation of a hyperplane. Two lines are said to be parallel if, given two distinct points P b Ql on the first line and P 2, Q2 on the second, the vectors and are parallel.
Two planes are said to be parallel in 3-space if their normal vectors are parallel. They are said to be perpendicular if their normal vectors are perpendicular. The angle between two planes is defined to be the angle between their normal vectors.
Find the cosine of the angle e between the planes. The equation of the plane through Q perpendicular to N is 2 X - Q.
The plane perpendicular to N, passing through P 1 is the desired plane. Distance between a point and a plane. We wish to find a formula for the distance between Q and the plane. By this we mean the length of the segment from Q to the point of intersection of the perpendicular line to the plane through Q, as on the figure.
We let Q' be this point of inter- section. Hence the distance between Q and the plane is 21 3. Write down vectors perpendicular to these lines. Find the equation of the line in 2-space, perpendicular to N and passing through P, for the following values of Nand P. Which of the following pairs of lines are perpendicular? Find the equation of the plane perpendicular to the given vector Nand passing through the given point P.
Find the equation of the plane passing through the following three points. Find a vector perpendicular to 1, 2, - 3 and 2, -1, 3 , and another vector perpendicular to - 1, 3, 2 and 2, 1, 1.
Find a parametric representation for the line of intersection of the planes of Exercises 10 and Find the point of the intersection of the line through P in the direction of N, and the plane through Q perpendicular to N.
Find the distance between the indicated point and plane. You have learned to solve such equations by the successive elimination of the variables. In this chapter, we shall review the theory of such equations, dealing with equations in n variables, and interpreting our results from the point of view of vectors.
Several geometric interpreta- tions for the solutions of the equations will be given. The first chapter is used here very little, and can be entirely omitted if you know only the definition of the dot product between two n-tuples. The multiplication of matrices will be formulated in terms of such a product. One geometric interpretation for the solutions of homogeneous equations will however rely on the fact that the dot product between two vectors is 0 if and only if the vectors are perpendicular, so if you are interested in this interpretation, you should refer to the section in Chapter I where this is explained.
Matrices We consider a new kind of object, matrices. An array of numbers all a l2 a l3 a ln a 2l a22 a23 a2n is called a matrix. The matrix has m rows and n columns. For instance, the first column is and the second row is a 2l , a 22 , We call aij the ij-entry or ij- component of the matrix.
The following is a 2 x 3 matrix: 1 4 The rows are 1,1, -2 and -1,4, The columns are Thus the rows of a matrix may be viewed as n-tuples, and the columns may be viewed as vertical m-tuples. A vertical m-tuple is also called a column vector. A column vector is an n x 1 matrix. When we write a matrix in the form aij , then i denotes the row and j denotes the column. A single number a may be viewed as a 1 x 1 matrix.
We note that we have met so far with the zero number, zero vector, and zero matrix. We shall now define addition of matrices and multiplication of ma- trices by numbers. We define addition of matrices only when they have the same size.
In other words, we add matrices of the same size componentwise. This is trivially verified. We shall now define the multiplication of a matrix by a number. We define cA to be the matrix whose ij-component is ca ij. We write Thus we multiply each component of A by c. Let A, B be as in Example 2. The matrix - A is also called the additive inverse of A. We define one more notion related to a matrix.
Taking the transpose of a matrix amounts to changing rows into columns and vice versa. Such a matrix is necessarily a square matrix. Remark on notation. I have written the transpose sign on the left, because in many situations one considers the inverse of a matrix written A-I, and then it is easier to write tA-I rather than A - 1 Y or At - 1, which are in fact equal. The mathematical community has no consensus as to where the transpose sign should be placed, on the right or left.
How do the diagonal elements of A and t A differ? Show that for any square matrix A, the matrix A - tA is skew-symmetric. Let Xl' Multiplication of Matrices We shall now define the product of matrices. Let Let A, B be as in Example 1. What do you find? Linear equations. Matrices give a convenient way of writing linear equations. You should already have considered systems of linear equations.
We shall see later how to solve such systems. We say that there are m equations and n unknowns, or n variables. Markov matrices. A matrix can often be used to represent a practical situation.
Suppose that any given year, some people leave each one of these cities to go to one of the others. The pelcentages of people leaving and going is given as follows, for each year. Ch goes to LA and t Ch goes to Bo. Such a matrix is called a Markov matrix.
If A is a square matrix, then we can form the product AA, which will be a square matrix of the same size as A. It is denoted by A2. Similarly, we can form A 3 , A 4 , and in general, An for any positive integer n. Thus An is the product of A with itself n times. We can define the unit n x n matrix to be the matrix having diagonal components all equal to 1, and all other components equal to O. You will find two different values. This is expressed by saying that mul- tiplication of matrices is not necessarily commutative.
For instance, powers of A commute, i. We now prove other basic properties of multiplication. Let A, B, C be matrices. Since our first assertion follows. As for the second, observe that the k-th column of xB is XBk. Since our second assertion follows. Associative law. Then A, BC can be multiplied. If we had started with the jl-component of BC and then computed the ii-component of A BC we would have found exactly the same sum, there by proving the desired property.
The above properties are very similar to those of multiplication of numbers, except that the commutative law does not hold. We can also relate multiplication with the transpose: Let A, B be matrices o. I t is occasion- ally convenient to rewrite the system in this fashion. Unlike division with non-zero numbers, we cannot divide by a matrix, any more than we could divide by a vector n-tuple. We do this only for square matrices.
Let A be an n x n matrix. Since we multiplied A with B on both sides, the only way this can make sense is if B is also an n x n matrix. Some matrices do not have in- verses.
However, if an inverse exists, then there is only one we say that the inverse is unique, or uniquely determined by A. This is easy to prove.
In other words, if B is a right in verse for A, then it is also a left in verse. You may assume this for the time being. Thus in verifying that a matrix is the in verse of another, you need only do so on one side.
We shall also find later a way of computing the inverse when it exists. It can be a tedious matter.
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