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Items 1 - 7 Linear algebra, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence, administrator of NASA. terney.info terney.info Textbook: Linear Algebra by Friedberg, Insel, Spence, 4th edition. Weekly schedule: . terney.info .pdf. New York: Pearson, p. This top-selling, theorem-proof book presents a careful treatment of the principle topics of linear algebra, and illustrates the.
Inner Products and Norms. The Adjoint of a Linear Operator. Normal and Self-Adjoint Operators. Unitary and Orthogonal Operators and Their Matrices. 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. The Jordan Canonical Form I. The Minimal Polynomial. Rational Canonical Form. Complex Numbers. Pearson offers special pricing when you package your text with other student resources.
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Out of print. Linear Algebra, 4th Edition. Stephen H. Spence, Illinois State University. Description For courses in Advanced Linear Algebra. NEW - Added section on the singular value decomposition. NEW - Revised proofs, added examples and exercises. Improves the clarity of the text and enhances students' understanding of it. The friendliest treatment of rigor in linear algebra —Usually used for a 2nd course, but can be used for smart, fast students in first course.
Numerous accessible exercises —Enriches and extends the text material. Real-world applications throughout. Reveals to students the power of the subject by demonstrating its practical uses. New to This Edition. Added section on the singular value decomposition. Revised proofs, added examples and exercises. Table of Contents 1. Share a link to All Resources. Instructor Resources. Previous editions. We encourage comments, which can be sent to us by e-mail or ordinary post.
Our web site and e-mail addresses are listed below. Friedberg Arnold J. Insel Lawrence R. Lor in this instance the motions of the swimmer and current.
For example. In most physical situations involving vectors. The magnitude of a velocity without. This geometry is derived from physical experiments that. Familiar situations suggest that when two like physical quantities act simultaneously at a point.
In tins section the geometry of vectors is discussed. Any such entity involving both magnitude and direction is called a "vector. Let o. The sum of two vectors x and y that act at the same point P is the vector beginning at P that is represented by the diagonal of parallelogram having x and y as adjacent sides.
If this is done. Then as Figure 1. Experiments show that if two like quantities act together. Z2 denote the endpoint of x and fei. This resultant vector is called the sum of the original vectors. Figure 1. The addition of vectors can be described algebraically with the use of analytic geometry. Since opposite sides of a parallelogram are parallel and of equal length.
Thus two vectors x and y that both act at the point P may be added "tail-to-head". In this case. Besides the operation of vector addition. See Figure 1. In the plane containing x and y. To describe scalar multiplication algebraically.
For all vectors x. Thus nonzero vectors having the same or opposite directions are parallel. These results can be used to write equations of lines and planes in space. For all vectors x and y. The algebraic descriptions of vector addition and scalar multiplication for vectors in a plane yield the following properties: The length of the arrow tx is i times the length of the arrow x.
If the endpoint of x has coordinates For each pair of real numbers a and 6 and each vector x. This operation. If the vector x is represented by an arrow. Consider first the equation of a line in space that passes through two distinct points A and B.
Let O denote the origin of a coordinate system in space, and let u and v denote the vectors that begin at. O and end at A and B, respectively. If w denotes the vector beginning at A and ending at B.
Since a scalar multiple of w is parallel to w but possibly of a different length than w, any point on the line joining A and B may be obtained as the endpoint of a vector that begins at A and has the form tw for some real number t. Conversely, the endpoint of every vector of the form tw that begins at A lies on the line joining A and B.
Notice also that the endpoint C of the vector v — u in Figure 1. Now let A, IS, and C denote any three noncollinear points in space. These points determine a unique plane, and its equation can be found by use of our previous observations about vectors.
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. A similar procedure locates the endpoint of tv.
Example 2 Let A, B, and C be the points having coordinates 1,0,2 , —3,-2,4 , and 1,8, -5 , respectively. Any mathematical structure possessing the eight properties on page 3 is called a vector space.
In the next section we formally define a vector space and consider many examples of vector spaces other than the ones mentioned above.
Determine whether the vectors emanating from the origin and terminating at the following pairs of points are parallel. Find the equations of the planes containing the following points in space. What arc the coordinates of the vector 0 in the Euclidean plane that satisfies property 3 on page 3? Justify your answer. Show that the midpoint of the line segment joining the points a. Prove that the diagonals of a parallelogram bisect each other. In Section 1. 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 field2 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. VS 5 For each clement x in V, 1. VS 6 For each pair of elements a, b in F and each element x in V, ab x — a bx.
The elements of the field F are called scalars and the elements of the vector space V are called vectors. The reader should not confuse this use of the word "vector'' with the physical entity discussed in Section 1. 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. Observe that VS 2 permits us to define the addition of any finite number of vectors unambiguously 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. The elements. Example 1 Thesel of all n-tuples with enl ries from a held F is denoted by F". Thus R'5 is n vector space over R. II and -5 1. C is a vector space over '. In this vector space. Since a 1 -1 uple whose only enl ry is from F can be regarded as an element of F, we usually write F rather than F1 for the vector space of l-tuples with entry from F.
The rows of the preceding matrix are regarded as vectors in F". The m - n matrix in which each entry equals zero is called the zero matrix and is denoted by. In this book, we denote matrices by capital italic letters e. B, and C. In addition, if the number of rows and columns of a matrix are equal.
Example 2 The set of all mxn matrices with entries from a Held F is a vector space, which we denote by M m x n F , with the following operations of matrix addition and scalar multiplication: The set J-'iS. Note that these are the familiar operations of addition and scalar multiplication for functions used in algebra and calculus. Note that the polynomials of degree zero may be written in the form f x — c for some nonzero scalar c.
Two polynomials. 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.
We will see in Exercise 23 of Section 2. Example 5 Let F be any field. A sequence in F is a function a from the positive integers into F. Our next two examples contain sets on which addition and scalar multiplication are defined, but which are not vector spaces.
Since VS f. VS 2 , and VS 8 fail to hold, S is not a vector space with these operations. Theorem 1. If x, y, and z are vectors in a vector space V. Thus x. Corollary 2. The vector y described in VS 4 is unique. The next result, contains some of the elementary properties of scalar multiplication. In any vector space V, the following statements are true: Label the following statements as true or false. A vector space may have more than one zero vector. A vector in F" may be regarded as a matrix in M.
An m x n matrix has m columns and n rows. In particular. Write the zero vector of 3. Perform the indicated operations. Wildlife Management.
Record the upstream and downstream crossings in two 3 x 3 matrices. At the end of May, a furniture1 store had the following inventory. Record these data as a 3 x 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?
In F S, R , show that. Prove Corollaries 1 and 2 of Theorem 1. Let V denote the set of all differentiablc 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. Let V denote the set of ordered pairs of real numbers. Is V a vector space over R with these operations?
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 x n matrices with real entries; so V is a vector space over R by Example 2. F be the field of rational numbers. Is V a vector space over F with the usual definitions of matrix addition and scalar multiplication? Define addition of elements of V coordinatewise. Is V a vector space over F with these operations?
V is a vector space over R. Because properties VS 1. Prove that. In any vector space V. Prove that Z is a vector space over F with the operations vi. See Example 5 for the definition of a sequence. Thus a subset W of a. The appropriate notion of substructure for vector spaces is introduced in this section. Fortunately it is not. See How many matrices arc there in the vector space M m x n Z2? Appendix C. The latter is called the zero subspace of V.
Let V and W be vector spaces over a field F. So condition a holds. Then W is a subspace ofV if and only if the following three conditions hold for the operations defined in V. I The preceding theorem provides a simple method for determining whether or not a given subset of a vector space is a subspace.
The transpose A1 of an m x n matrix A is the n x m matrix obtained from A by interchanging the rows with the columns: W has a zero vector. Each vector in W has an additive inverse in W. I"he zero matrix is equal to its transpose and hence belongs to W. Hence W is a subspace of V. See Exercise 3. Using this fact. W is closed under scalar multiplication. It is easily proved that for any matrices A and B and any scalars a and 6. The set W of all symmetric matrices " u: Let V be a vector space and W a subset ofV.
The next theorem shows that the zero vector of W must be the same as the zero vector of V ami thai property I is redundant. Hence conditions! W is closed under addition. If W is a subspace of V. R defined in Example 3 of Section 1. The first. Example 1 Let it be a nonnegative integer. Clearly the1 zero matrix is a diagonal matrix because all of its emtric's are1 0.
Since constant functions are1 continuous.. First note that the1 zero of 1F R. Clearly C 7? Since the zc. It therefore follows from Theorem 1. So C R is closed under addition and scalar multiplication and hene-e1 is a subspace of J- R. The example's that follow provide further illustrations of the concept of a subspace. R by Theorem 1. If V is a vector space and W is a subset of V that is a vector space. It is easily seen that the. Hence X f y and a.
In fact. This idea is explored in Exercise Let c. See Exercise There is. Let C be a collection of subspaces of V. As we already have suggested. Then x and y are' contained in each subspace in C.
I Having shown that the intersection of subspaces of a vector space V is a subspace of V. In addition. Determine the transpose of each of the matrices that follow. Describe Wi n W 3. Let Wi. Prove that diagonal matrices are symmetric matrices. Justify your answers. Determine whether the following sets are subspaces of R3 under the operations of addition and scalar multiplication defined on R3.
Prove that a subset W of a vector space V is a subspace. Prove that a subset W of a vector space. Prove that Cn R is a subspace of F R. Let Cn R. Prove that C S. An m x n matrix A is called upper triangular if all entries lying below the diagonal entries are zero. Is the set. Let S be a nonempty set and F a.
F such that f s — 0 for all but a finite number of elements of S. Is the set of all differentiable real-valued functions defined on R a subspace of C R? Let W. Prove that W.. Let C S. F2 are subspace. F n is the direct sum of the subspaces W.
Let V demote the vector space consisting of all upper triangular n x n matrices as defined in Exercise Show that. Let Wi and W 2 be subspaces of a vector space V. Prove that the set Wi of all skew-symmetric n x n matrices with entries from F is a subspae. Define W. Let W be a subspace of a vector space V over a field F.
Now assume that F is not of characteristic 2 see Appendix C. Let F be a. It is customary to denote this coset. Compare this exercise with Exercise Let F be a field that is not of characteristic 2.
Let Wi and W2 be subspaces of a vector space V. Prove that V is the direct sum of Wi and W 2 if and only if each vector in V can be uniquely written as x.
This is proved as Theorem 1. Department of Agriculture. Composition of Foods Agriculture Handbook Number 8. Consumer and food Economics Research Division. Thus the zero vector is a. Rernice K. Let V be a vector space and S a nonempty subset ofV.
A and ending at B and C. An important special case occurs when A is the origin.
Observe that in any vector space V. Watt and Annabel I.. The appropriate. So grams of cupcake. Chapters 1 and 2 we encounter many different situations in which it is necessary to determine1 whether or not a vector can be expressed as a linear combination of other vectors. B 2 riboflavin. The vitamin content of grams of each food can be recorded as a column vector in R: For now.
In Chapter 3. OOX 0. Bi thiamine. We now want to make the coefficient of 03 in the second equation equal to 1. To do this. 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. To begin. The result is the following new system: Hence 2. Thus we must determine if there are scalars Next we add —3 times the second equation to the third.
To solve system I. In Section 3. The first nonzero coefficient in each equation is one. The procedure: It is easy to solve for the first unknown present in each of the equations If an unknown is the first. Thus for any choice of scalars a 2 and a. Rewriting system 4 in this form. Therefore 2. The first unknown with a nonzero coefficient in any equation has a larger subscript than the first unknown with a nonzc: Note that we employed these operations to obtain a system of equations that had the following properties: Sex1 Example 2.
We return to the study of systems of linear equations in Chapter 3. Example 2 We claim that 2x3. In the first. We diseniss there the theoretical basis for this method of solving systems of linear equations and further simplify the procedure by use of matrices. Using the preceding technique. Eliminating a. Thus we are led to the following system of linear equations: In the second case. The span of any subset S of a vector space V is a subspace of V.
Let S be a nonempty subset of a vector space V. Thus span S is a subspace of V. If w C span. We now name such a. For convenience. Since S C W. In R3. The span of S. Because iu.
Proof This result is immediate if. Now let W denote any subspace of V that contains S. This fact is true in general. R c a n be expressed as a linear combination of the four given matrices as follows: 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. So any linear combination of these matrices has equal diagonal entries.
Hence not every 2 x 2 matrix is a linear combination of these three matrices. Thus x G R3 is a linear combination of u. It is natural to seek a subset of W that generates W and is as small as possible. Solve the following systems of linear equations by the method introduced in this section. For each of the following lists of vectors in R3. For each list of polynomials in P 3 K. In each part. Show that if Si and S2 are arbitrary subsets of a vector space V.
Show that the vectors 1. The sum of two subsets is defined in the exercises of Section 1. Show that the matrices 1 0 0 0 generate M2X2 F Interpret this result geometrically in R3.
Let W be a subspace of a vector space V. The reader should verify that no such solution exists. Thus U4 is a linear combination of ui. It can be shown. This does not. Unless W is the zero subspace.
Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S. Let us attempt to find a proper subset of S that also generates W. Give an example in which span Si D S 2 and span Si nspan S 2 are equal and one in which they are unequal.
It is desirable to find a "small" finite subset S that generates W because we can then describe each vector in W as a linear combination of the finite number of vectors in S. Under what conditions are there only a finite number of distinct subsets S of W such that S 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. W is an infinite set. For any vectors Wi. In this case we also say that the vectors of S are linearly dependent. To show that. For instance. A subset S of a vector space V is called linearly dependent if there exist a finite number of distinct vectors W].
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. That is. 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. By formulating our question differently. This observation leads us to the following definition. As before. Finding such scalars amounts to finding a nonzero solution to the system of linear equations ai 4.
A subset S of a vector space that is not linearly dependent is called linearly independent. The empty set is linearly independent.
Example 2 In M 2x3 i? A set is linearly independent if and only if the only representations of 0 as linear combinations of its vectors are trivial representations. A set consisting of a single nonzero vector is linearly independent. The following facts about linearly independent sets are true in any vector space.
Thus S is a linearly dependent subset of R4. For if a0p0 x 4. Suppose that ai. Equating the corresponding coordinates of the vectors on the left and the right sides of this equation. This technique is illustrated in the examples that follow.
It follows that if no proper subset of S generates the span of S. This equation implies that w3 or alternatively.
S S I Corollary. OM4 4. 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. Let S be a linearly independent subset of a vector space V. More generally. We have previously noted that S is linearly dependent: If S 2 is linearly independent. Another way to view the preceding statement is given in Theorem 1. Let V be a vector space.
To see why. If Si is linearly dependent. I Earlier in this section. Then there exist vectors Vi.
Thus aiv 4. H Linearly independent generating sets are investigated in detail in Section 1. Since v is a linear combination of u2. Because S is linearly independent. Determine whether the following sets are linearly dependent or linearly independent.
Recall from Example 3 in Section 1. Find a linearly independent set that generates this subspace. Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another.
Let M be a square upper triangular matrix as defined in Exercise 12 of Section 1. Let S be a set of nonzero polynomials in P F such that no two have the same degree. Prove Theorem 1. Let V be a vector space over a field of characteristic not equal to two.
This property is proved below in Theorem Prove that the columns of M are linearly independent. Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent. 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. It is this property that makes linearly independent generating sets the building blocks of vector spaces.
Prove that S is linearly independent. How many vectors are there in span S? Hence not every vector space has a finite basis. Let ft be a basis for V. Tims v is a linear combination of the vectors of ft. The next theorem. A basis j3 for a vector space V is a linearly independent subset of V that generates V.
If ft is a basis for V. Then ft is a basis for V if and only if each o G V can be uniquely expressed as a linear combination of vectors of ft. Uk is a linearly independent subset of S. This method is illustrated in the next example. Thus v determines a unique n-tuple of scalars a i. Since ft is linearly independent.. Since S is a finite set. Hence V has a finite basis. The proof of the converse is an exercise. Because ft is linearly independent by construction.
We see in Section 2. By item 2 on page We claim that ft is a basis for V. If a vector space V is generated by a Bnite set S. I Theorem 1. This fact suggests that V is like the vector space F n.. In this book. By Theorem 1. I Because of the method by which the basis ft was obtained in the proof of Theorem Hence we do not include 8.
Since 4 2. Thus we include 1. Let V be a vector space vectors. An easy calculation shows that this set is linearly independent. Because 2 2. The proof is by mathematical induction on m. On the other hand.
We can select a basis for R3 that is a subset of S by the technique used in proving Theorem 1. To start. Since S is linearly independent and ft generates V. We prove that the theorem is true for m Let V be a vector space having a tinitc basis Corollary 1 asserts that the number of vectors in any basis for V is an intrinsic property of V.
This completes the induction.. Therefore 7 is finite. If 7 contains more than n vectors.. Then every basis for V contains the same number of vectors.. Theorem The unique number of vectors. This fact makes possible the following important definitions A vector space is called finite-dimensional if it has a basis consisting of a Unite number of vectors. By the corollary to Theorem 1. I If a vector space has a finite basis. Reversing the roles of ft and 7 and arguing as above.
Suppose that ft is a finite basis for V that contains exactly n vectors. In Section From this fact it follows that the vector space P F is infinite-dimensional because it has an infinite linearly independent set. A vector space that is not finite-dimensional is called infinite-dimensional.
Just as no linearly independent subset of a finite-dimensional vector space V can contain more than dim V vectors. The following results are consequences of Examples 1 through 4. Let V be a vector space with dimension n. Example 8 The vector space F n has dimension n.
Example 9 The vector space M m x n F has dimension rnn. This set is. Yet nothing that we have proved in this section guarantees an infinite-dimensional vector space must have a basis G must contain at least n vectors. It follows from Example 3 of Section 1.
Since L is also linearly independent. L is a basis for V. Now L U H contains at most n vectors. Corollary 1 implies that H contains exactly n vectors. Since a subset of G contains n vectors. This procedure also can be used to extend a linearly independent set to a basis. It follows from Example 4 A procedure for reducing a generating set to a basis was illustrated in Example 6.
Thus if the dimension of V is n. The Venn diagram in Figure 1. If V has a finite basis. For this reason. A basis for a vector space V is a linearly independent subset of V that generates V. This number is called the dimension of V. W contains a nonzero vector Xj. Since no linearly independent subset of V can contain more than n vectors. Let W be a subspace of a finite-dimensional vector space V.
Continue choosing vectors. But Corollary 2 of the replacement theorem implies that this basis for W is also a basis for V. Cn n Ci. Any subspace of R2 having dimension 1 consists of all scalar multiples of some nonzero vector in R2 Exercise 11 of Section 1. If W is a subspace of a finite-dimensional vector space V. The polynomials fo x. A subspace of R2 having dimension 0 consists of the origin of the Euclidean plane.
Because S is a linearly independent. The Lagrange Interpolation Formula Corollary 2 of the replacement theorem can be applied to obtain a useful formula. If a point of R2 is identified in the natural way with a point in the Euclidean plane. Interpreting these possibilities geometrically.. Let Cq. Corollary 2 of the replacement theorem guarantees that S can be extended to a basis for V. J Example 20 The set of all polynomials of the form a18xla16x Let S be a basis for W. Note that each fi x is a polynomial of degree n and hence is in P n F.
This representation is called the Lagrange interpolation formula. Because ft is a basis for P n F. Every vector space has a finite basis. Every vector space that is generated by a finite set has a basis. An important consequence of the Lagrange interpolation formula is the following result: The Lagrange polynomials associated with Co.
A vector space cannot have more than one basis. For example.. Do the polynomials x 3 — 2x 2 4T. The vectors u: Give three different bases for F 2 and for M 2X 2 F. Determine which of the following sets are bases for R3. Then Si cannot contain more vectors than S 2. Determine which of the following sets are bases for P2 R.
Let u. Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to zero. The set of solutions to the system of linear equations xi. Find a basis for this subspace. What are the dimensions of Wi and W2? Find bases for the following subspaces of F 5: Let u and v be distinct vectors of a vector space V. Find the unique representation of an arbitrary vector Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset.
Let f x be a polynomial of degree n in Pn R.. What is the dimension of W? Let Vi. Find a basis for W. Let V. Be careful not to assume that S is finite If V and W are vector spaces over F of dimensions m and n.
The set of all n x n matrices having trace equal to zero is a subspace W Exam of M n x n F see Example 4 of Section 1. Find a basis for the vector space in Example 5 of Section 1.