An Introduction to Linear AlgebraCourier Corporation, 3 déc. 2012 - 464 pages "The straight-forward clarity of the writing is admirable." — American Mathematical Monthly. This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra. Although its accessibility makes it suitable for non-mathematicians, Professor Mirsky's book is nevertheless a systematic and rigorous development of the subject. Part I deals with determinants, vector spaces, matrices, linear equations, and the representation of linear operators by matrices. Part II begins with the introduction of the characteristic equation and goes on to discuss unitary matrices, linear groups, functions of matrices, and diagonal and triangular canonical forms. Part II is concerned with quadratic forms and related concepts. Applications to geometry are stressed throughout; and such topics as rotation, reduction of quadrics to principal axes, and classification of quadrics are treated in some detail. An account of most of the elementary inequalities arising in the theory of matrices is also included. Among the most valuable features of the book are the numerous examples and problems at the end of each chapter, carefully selected to clarify points made in the text. |
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Page 23
... find it helpful actually to carry out the procedure described here, say for the case n = 4, k=2,r1—_- l,r2=3. As an illustration of the use of Laplace's expansion we shall evaluate the determinant 0 “I3 “14 “I5 0 0 a23 a24 0 0 a33 0 0 0 ...
... find it helpful actually to carry out the procedure described here, say for the case n = 4, k=2,r1—_- l,r2=3. As an illustration of the use of Laplace's expansion we shall evaluate the determinant 0 “I3 “14 “I5 0 0 a23 a24 0 0 a33 0 0 0 ...
Page 34
... Find the discriminant of the equation 2:'-1-aa:+b = 0. Hence show that the equation 2:'-|—4:c+ 3 = 0 has at least two coincident roots. 6. Ifs, = a'+B'+y" and show that = 4a'b'c'. 8n 8n+1 80+! 8n+1 8n+l 8n+3 8n+2 8n+3 an-H A,,= show ...
... Find the discriminant of the equation 2:'-1-aa:+b = 0. Hence show that the equation 2:'-|—4:c+ 3 = 0 has at least two coincident roots. 6. Ifs, = a'+B'+y" and show that = 4a'b'c'. 8n 8n+1 80+! 8n+1 8n+l 8n+3 8n+2 8n+3 an-H A,,= show ...
Page 35
... ama 0 Oaxa 0 0 aa: Show how to find the value of D,,; and prove that, if a = 1, it = 2cos0 (0 < 0 < 'n-),thenD,, = sin(n+1)0/sinfl. 13. Prove that I+a1I a2 a3 . . . . a,I a1 1+a2 a3 . . . . a,, a1 a2 l+a3 . . . .a,, = 1+a,+a2+a,+...+a, ...
... ama 0 Oaxa 0 0 aa: Show how to find the value of D,,; and prove that, if a = 1, it = 2cos0 (0 < 0 < 'n-),thenD,, = sin(n+1)0/sinfl. 13. Prove that I+a1I a2 a3 . . . . a,I a1 1+a2 a3 . . . . a,, a1 a2 l+a3 . . . .a,, = 1+a,+a2+a,+...+a, ...
Page 38
... . - . 1<|-<|<,| 1 an 04 . . . 04:'2 04,," 31. Use the result of Problem 21 to evaluate the n-rowed determinant a — l — 1 . . . — 1 —la—l...—1 —1—l—l....a II VECTOR SPACES AND LINEAR MANIFOLDS WE shall find it. 38 DETERMINANTS I.
... . - . 1<|-<|<,| 1 an 04 . . . 04:'2 04,," 31. Use the result of Problem 21 to evaluate the n-rowed determinant a — l — 1 . . . — 1 —la—l...—1 —1—l—l....a II VECTOR SPACES AND LINEAR MANIFOLDS WE shall find it. 38 DETERMINANTS I.
Page 39
L. Mirsky. II VECTOR SPACES AND LINEAR MANIFOLDS WE shall find it convenient to use the customary symbolism to denote class membership, writing 2: E X to indicate that an object 2: is a member of a class X and z ¢ X to indicate that it ...
L. Mirsky. II VECTOR SPACES AND LINEAR MANIFOLDS WE shall find it convenient to use the customary symbolism to denote class membership, writing 2: E X to indicate that an object 2: is a member of a class X and z ¢ X to indicate that it ...
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A₁ algebra assertion automorphism b₁ basis bilinear form bilinear operator canonical forms characteristic roots characteristic vectors coefficients columns commute complex numbers convergent coordinates Deduce defined denote determinant diagonal form diagonal matrix dimensionality E-operations edition equal equivalence EXERCISE exists follows functions geometry given Hence hermitian form hermitian matrix identity implies inequality integers invariant space isomorphic linear equations linear manifold linear transformation linearly independent matrix group minimum polynomial multiplication non-singular linear transformation non-singular matrix non-zero numbers nxn matrix obtain orthogonal matrix positive definite positive semi-definite possesses problems proof of Theorem prove quadratic form quadric rank relation represented respect result rotation S-¹AS satisfies scalar Show similar singular skew-symmetric matrix solution square matrix suppose symmetric matrix t₁ theory tions triangular unique unit element unitary matrix values vector space view of Theorem write x₁ xTAx y₁ zero