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{{InfoboxBook | {{InfoboxBook | ||
|title=Calculus | |title=Calculus | ||
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|isbn13=978-0471000051 | |isbn13=978-0471000051 | ||
}} | }} | ||
The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. | {{NavContainerFlex | ||
|content= | |||
{{NavButton|link=[[Read#Basic_Mathematics|Read]]}} | |||
}} | |||
The textbook [https://simeioseismathimatikwn.files.wordpress.com/2013/03/apostol-calculusi.pdf '''''Calculus'''''] by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. It provides a rigorous treatment of theory and application, in addition to the historical context of its topics. It should be noted that there is a [https://archive.org/details/calculus-tom-m.-apostol-calculus-volume-2-2nd-edition-proper-2-1975-wiley-sons-libgen.lc/Apostol%20T.%20M.%20-%20Calculus%20vol%20II%20%281967%29/ second volume], not listed here, which covers multivariable topics and applications to subjects such as probability. | |||
== Table of Contents == | == Table of Contents == | ||
Line 109: | Line 113: | ||
| 1.2 || Functions. Informal description and examples || 50 | | 1.2 || Functions. Informal description and examples || 50 | ||
|- | |- | ||
| 1.3 || Functions. Formal definition as a set of ordered pairs || 53 | | <nowiki>*</nowiki>1.3 || Functions. Formal definition as a set of ordered pairs || 53 | ||
|- | |- | ||
| 1.4 || More examples of real functions || 54 | | 1.4 || More examples of real functions || 54 | ||
Line 149: | Line 153: | ||
| 1.22 || Calculation of the integral of a bounded monotonic function || 79 | | 1.22 || Calculation of the integral of a bounded monotonic function || 79 | ||
|- | |- | ||
| 1.23 || Calculation of the integral | | 1.23 || Calculation of the integral <math>\int_0^b x^p dx</math> when <math>p</math> is a positive integer || 79 | ||
|- | |- | ||
| 1.24 || The basic properties of the integral || 80 | | 1.24 || The basic properties of the integral || 80 | ||
Line 159: | Line 163: | ||
| 1.27 || Proofs of the basic properties of the integral || 84 | | 1.27 || Proofs of the basic properties of the integral || 84 | ||
|- | |- | ||
! colspan="3" | | ! colspan="3" | 2. SOME APPLICATIONS OF INTEGRATION | ||
|- | |||
| 2.1 || Introduction || 88 | |||
|- | |||
| 2.2 || The area of a region between two graphs expressed as an integral || 88 | |||
|- | |||
| 2.3 || Worked examples || 89 | |||
|- | |||
| 2.4 || Exercises || 94 | |||
|- | |||
| 2.5 || The trigonometric functions || 94 | |||
|- | |||
| 2.6 || Integration formulas for the sine and cosine || 94 | |||
|- | |||
| 2.7 || A geometric description of the sine and cosine functions || 94 | |||
|- | |||
| 2.8 || Exercises || 94 | |||
|- | |||
| 2.9 || Polar coordinates || 94 | |||
|- | |||
| 2.10 || The integral for area in polar coordinates || 94 | |||
|- | |||
| 2.11 || Exercises || 94 | |||
|- | |||
| 2.12 || Application of integration to the calculation of volume || 94 | |||
|- | |||
| 2.13 || Exercises || 94 | |||
|- | |||
| 2.14 || Application of integration to the calculation of work || 94 | |||
|- | |||
| 2.15 || Exercises || 94 | |||
|- | |||
| 2.16 || Average value of a function || 94 | |||
|- | |||
| 2.17 || Exercises || 94 | |||
|- | |||
| 2.18 || The integral as a function of the upper limit. Indefinite integrals || 94 | |||
|- | |||
| 2.19 || Exercises || 94 | |||
|- | |||
! colspan="3" | 3. CONTINUOUS FUNCTIONS | |||
|- | |||
| 3.1 || Informal description of continuity || 126 | |||
|- | |||
| 3.2 || The definition of the limit of a function || 127 | |||
|- | |||
| 3.3 || The definition of continuity of a function || 130 | |||
|- | |||
| 3.4 || The basic limit theorems. More examples of continuous functions || 131 | |||
|- | |||
| 3.5 || Proofs of the basic limit theorems || 135 | |||
|- | |||
| 3.6 || Exercises || 138 | |||
|- | |||
| 3.7 || Composite functions and continuity || 140 | |||
|- | |||
| 3.8 || Exercises || 142 | |||
|- | |||
| 3.9 || Bolzano's theorem for continuous functions || 142 | |||
|- | |||
| 3.10 || The intermediate-value theorem for continuous functions || 144 | |||
|- | |||
| 3.11 || Exercises || 145 | |||
|- | |||
| 3.12 || The process of inversion || 146 | |||
|- | |||
| 3.13 || Properties of functions preserved by inversion || 147 | |||
|- | |||
| 3.14 || Inverses of piecewise monotonic functions || 148 | |||
|- | |||
| 3.15 || Exercises || 149 | |||
|- | |||
| 3.16 || The extreme-value theorem for continuous functions || 150 | |||
|- | |||
| 3.17 || The small-span theorem for continuous functions (uniform continuity) || 152 | |||
|- | |||
| 3.18 || The integrability theorem for continuous functions || 152 | |||
|- | |||
| 3.19 || Mean-value theorems for integrals of continuous functions || 154 | |||
|- | |||
| 3.20 || Exercises || 155 | |||
|- | |||
! colspan="3" | 4. DIFFERENTIAL CALCULUS | |||
|- | |||
| 4.1 || Historical introduction || 156 | |||
|- | |||
| 4.2 || A problem involving velocity || 157 | |||
|- | |||
| 4.3 || The derivative of a function || 159 | |||
|- | |||
| 4.4 || Examples of derivatives || 161 | |||
|- | |||
| 4.5 || The algebra of derivatives || 164 | |||
|- | |||
| 4.6 || Exercises || 167 | |||
|- | |||
| 4.7 || Geometric interpretation of the derivative as a slope || 169 | |||
|- | |||
| 4.8 || Other notations for derivatives || 171 | |||
|- | |||
| 4.9 || Exercises || 173 | |||
|- | |||
| 4.10 || The chain rule for differentiating composite functions || 174 | |||
|- | |||
| 4.11 || Applications of the chain rule. Related rates and implicit differentiation || 176 | |||
|- | |||
| 4.12 || Exercises || 179 | |||
|- | |||
| 4.13 || Applications of the differentiation to extreme values of cuntions|| 181 | |||
|- | |||
| 4.14 || The mean-value theorem for derivatives || 183 | |||
|- | |||
| 4.15 || Exercises || 186 | |||
|- | |||
| 4.16 || Applications of the mean-value theorem to geometric properties of functions || 187 | |||
|- | |||
| 4.17 || Second-derivative test for extrema || 188 | |||
|- | |||
| 4.18 || Curve sketching || 189 | |||
|- | |||
| 4.19 || Exercises || 191 | |||
|- | |||
| 4.20 || Worked examples of extremum problems || 191 | |||
|- | |||
| 4.21 || Exercises || 194 | |||
|- | |||
| <nowiki>*</nowiki>4.22 || Partial derivatives || 196 | |||
|- | |||
| <nowiki>*</nowiki>4.23 || Exercises || 201 | |||
|- | |||
! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION | |||
|- | |||
| 5.1 || The derivative of an indefinite integral. The first fundamental theorem of calculus || 202 | |||
|- | |||
| 5.2 || The zero-derivative theorem || 204 | |||
|- | |||
| 5.3 || Primitive functions and the second fundamental theorem of calculus || 205 | |||
|- | |||
| 5.4 || Properties of a function deduced from properties of its derivative || 207 | |||
|- | |||
| 5.5 || Exercises || 208 | |||
|- | |||
| 5.6 || The Leibniz notation for primitives || 210 | |||
|- | |||
| 5.7 || Integration by substitution || 212 | |||
|- | |||
| 5.8 || Exercises || 216 | |||
|- | |||
| 5.9 || Integration by parts || 217 | |||
|- | |||
| 5.10 || Exercises || 220 | |||
|- | |||
| <nowiki>*</nowiki>5.11 || Miscellaneous review exercises || 222 | |||
|- | |||
! colspan="3" | 6. THE LOGARITHM, THE EXPONENTIAL, AND THE INVERSE TRIGONOMETRIC FUNCTIONS | |||
|- | |||
| 6.1 || Introduction || 226 | |||
|- | |||
| 6.2 || Motivation for the definition of the natural logarithm as an integral || 227 | |||
|- | |||
| 6.3 || The definition of the logarithm. Basic properties || 229 | |||
|- | |||
| 6.4 || The graph of the natural logarithm || 230 | |||
|- | |||
| 6.5 || Consequences of the functional equation <math>L(ab) = L(a) + L(b)</math> || 230 | |||
|- | |||
| 6.6 || Logarithms referred to any positive base <math>b \ne 1</math> || 232 | |||
|- | |||
| 6.7 || Differentiation and integration formulas involving logarithms || 233 | |||
|- | |||
| 6.8 || Logarithmic differentiation || 235 | |||
|- | |||
| 6.9 || Exercises || 236 | |||
|- | |||
| 6.10 || Polynomial approximations to the logarithm || 236 | |||
|- | |||
| 6.11 || Exercises || 242 | |||
|- | |||
| 6.12 || The exponential function || 242 | |||
|- | |||
| 6.13 || Exponentials expressed as powers of e || 242 | |||
|- | |||
| 6.14 || The definition of <math>e^x</math> for arbitrary real x || 244 | |||
|- | |||
| 6.15 || The definition of <math>a^x</math> for <math>a > 0</math> and x real || 245 | |||
|- | |||
| 6.16 || Differentiation and integration formulas involving exponentials || 245 | |||
|- | |||
| 6.17 || Exercises || 248 | |||
|- | |||
| 6.18 || The hyperbolic functions || 251 | |||
|- | |||
| 6.19 || Exercises || 251 | |||
|- | |||
| 6.20 || Derivatives of inverse functions || 252 | |||
|- | |||
| 6.21 || Inverses of the trigonometric functions || 253 | |||
|- | |||
| 6.22 || Exercises || 256 | |||
|- | |||
| 6.23 || Integration by partial fractions || 258 | |||
|- | |||
| 6.24 || Integrals which can be transformed into integrals of rational functions || 264 | |||
|- | |||
| 6.25 || Exercises || 267 | |||
|- | |||
| 6.26 || Miscellaneous review exercises || 268 | |||
|- | |||
! colspan="3" | 7. POLYNOMIAL APPROXIMATIONS TO FUNCTIONS | |||
|- | |||
| 7.1 || Introduction || 272 | |||
|- | |- | ||
| 7.2 || The Taylor polynomials generated by a function || 273 | |||
|- | |- | ||
| | | 7.3 || Calculus of Taylor polynomials || 275 | ||
|- | |- | ||
| | | 7.4 || Exercises || 278 | ||
|- | |- | ||
| | | 7.5 || Taylor's formula with remainder || 278 | ||
|- | |- | ||
| 7.6 || Estimates for the error in Taylor's formula || 280 | |||
|- | |- | ||
| | | <nowiki>*</nowiki>7.7 || Other forms of the remainder in Taylor's formula || 283 | ||
|- | |- | ||
| | | 7.8 || Exercises || 284 | ||
|- | |- | ||
| | | 7.9 || Further remarks on the error in Taylor's formula. The o-notation || 286 | ||
|- | |- | ||
| | | 7.10 || Applications to indeterminate forms || 289 | ||
|- | |- | ||
| | | 7.11 || Exercises || 290 | ||
|- | |- | ||
| | | 7.12 || L'Hopital's rule for the indeterminate form 0/0 || 292 | ||
|- | |- | ||
| 7.13 || Exercises || 295 | |||
|- | |- | ||
| | | 7.14 || The symbols <math>+\inf</math> and <math>-\inf</math>. Extension of L'Hopital's rule || 296 | ||
|- | |- | ||
| | | 7.15 || Infinite limits || 298 | ||
|- | |- | ||
| 7.16 || The behavior of log<math>x</math> and <math>e^x</math> for large <math>x</math> || 300 | |||
|- | |- | ||
| 7.17 || Exercises || 303 | |||
|- | |- | ||
| | ! colspan="3" | 8. INTRODUCTION TO DIFFERENTIAL EQUATIONS | ||
|- | |- | ||
| | | 8.1 || Introduction || 305 | ||
|- | |- | ||
| | | 8.2 || Terminology and notation || 306 | ||
|- | |- | ||
| | | 8.3 || A first-order differential equation for the exponential function || 307 | ||
|- | |- | ||
| 8.4 || First-order linear differential equations || 308 | |||
|- | |- | ||
| | | 8.5 || Exercises || 311 | ||
|- | |- | ||
| | | 8.6 || Some physical problems leading to first-order linear differential equations || 313 | ||
|- | |- | ||
| 8.7 || Exercises || 319 | |||
|- | |- | ||
| | | 8.8 || Linear equations of second order with constant coefficients || 322 | ||
|- | |- | ||
| | | 8.9 || Existence of solutions of the equation <math>y^{''} + by = 0</math> || 323 | ||
|- | |- | ||
| | | 8.10 || Reduction of the general equation to the special case <math>y^{''} + by = 0</math> || 324 | ||
|- | |- | ||
| | | 8.11 || Uniqueness theorem for the equation <math>y^{''} + by = 0</math> || 324 | ||
|- | |- | ||
| 8.12 || Complete solution of the equation <math>y^{''} + by = 0</math> || 326 | |||
|- | |- | ||
| | | 8.13 || Complete solution of the equation <math>y^{''} + ay^{'} + by = 0</math> || 326 | ||
|- | |- | ||
| | | 8.14 || Exercises || 328 | ||
|- | |- | ||
| | | 8.15 || Nonhomogeneous linear equations of second order with constant coefficients || 329 | ||
|- | |- | ||
| | | 8.16 || Special methods for determining a particular solution of the nonhomogeneous equation <math>y^{''} + ay^{'} + by = R</math> || 332 | ||
|- | |- | ||
| | | 8.17 || Exercises || 333 | ||
|- | |- | ||
| | | 8.18 || Examples of physical problems leading to linear second-order equations with constant coefficients || 334 | ||
|- | |- | ||
| 8.19 || Exercises || 339 | |||
|- | |- | ||
| | | 8.20 || Remarks concerning nonlinear differential equations || 339 | ||
|- | |- | ||
| | | 8.21 || Integral curves and direction fields || 341 | ||
|- | |- | ||
| | | 8.22 || Exercises || 344 | ||
|- | |- | ||
| | | 8.23 || First-order separable equations || 345 | ||
|- | |- | ||
| | | 8.24 || Exercises || 347 | ||
|- | |- | ||
| 8.25 || Homogeneous first-order equations || 347 | |||
|- | |- | ||
| 8.26 || Exercises || 350 | |||
|- | |- | ||
| | | 8.27 || Some geometrical and physical problems leading to first-order equations || 351 | ||
|- | |- | ||
| | | 8.28 || Miscellaneous review exercises || 355 | ||
|- | |- | ||
! colspan="3" | 9. COMPLEX NUMBERS | |||
|- | |- | ||
| | | 9.1 || Historical introduction || 358 | ||
|- | |- | ||
| | | 9.2 || Definitions and field properties || 358 | ||
|- | |- | ||
| 9.3 || The complex numbers as an extension of the real numbers || 360 | |||
|- | |- | ||
| | | 9.4 || The imaginary unit <math>i</math> || 361 | ||
|- | |- | ||
| | | 9.5 || Geometric interpretation. Modulus and argument || 362 | ||
|- | |- | ||
| | | 9.6 || Exercises || 365 | ||
|- | |- | ||
| 9.7 || Complex exponentials || 366 | |||
|- | |- | ||
| | | 9.8 || Complex-valued functions || 368 | ||
|- | |- | ||
| | | 9.9 || Examples of differentiation and integration formulas || 369 | ||
|- | |- | ||
| 9.10 || Exercises || 371 | |||
|- | |- | ||
| | ! colspan="3" | 10. SEQUENCES, INFINITE SERIES, IMPROPER INTEGRALS | ||
|- | |- | ||
| | | 10.1 || Zeno's paradox || 374 | ||
|- | |- | ||
| | | 10.2 || Sequences || 378 | ||
|- | |- | ||
| 10.3 || Monotonic sequences of real numbers || 381 | |||
|- | |- | ||
| | | 10.4 || Exercises || 382 | ||
|- | |- | ||
| | | 10.5 || Infinite series || 383 | ||
|- | |- | ||
| | | 10.6 || The linearity property of convergent series || 385 | ||
|- | |- | ||
| | | 10.7 || Telescoping series || 386 | ||
|- | |- | ||
| | | 10.8 || The geometric series || 388 | ||
|- | |- | ||
| | | 10.9 || Exercises || 391 | ||
|- | |- | ||
! colspan="2" | Index || | | <nowiki>*</nowiki>10.10 || Exercises on decimal expansions || 393 | ||
|- | |||
| 10.11 || Tests for convergence || 394 | |||
|- | |||
| 10.12 || Comparison tests for series of nonnegative terms || 394 | |||
|- | |||
| 10.13 || The integral test || 397 | |||
|- | |||
| 10.14 || Exercises || 398 | |||
|- | |||
| 10.15 || The root test and the ratio test for series of nonnegative terms || 399 | |||
|- | |||
| 10.16 || Exercises || 402 | |||
|- | |||
| 10.17 || Alternating series || 403 | |||
|- | |||
| 10.18 || Conditional and absolute convergence || 406 | |||
|- | |||
| 10.19 || The convergence tests of Dirichlet and Abel || 407 | |||
|- | |||
| 10.20 || Exercises || 409 | |||
|- | |||
| <nowiki>*</nowiki>10.21 || Rearrangements of series || 411 | |||
|- | |||
| 10.22 || Miscellaneous review exercises || 414 | |||
|- | |||
| 10.23 || Improper integrals || 416 | |||
|- | |||
| 10.24 || Exercises || 420 | |||
|- | |||
! colspan="3" | 11. SEQUENCES AND SERIES OF FUNCTIONS | |||
|- | |||
| 11.1 || Pointwise convergence of sequences of functions || 422 | |||
|- | |||
| 11.2 || Uniform convergence of sequences of functions || 423 | |||
|- | |||
| 11.3 || Uniform convergence and continuity || 424 | |||
|- | |||
| 11.4 || Uniform convergence and integration || 425 | |||
|- | |||
| 11.5 || A sufficient condition for uniform convergence || 427 | |||
|- | |||
| 11.6 || Power series. Circle of convergence || 428 | |||
|- | |||
| 11.7 || Exercises || 430 | |||
|- | |||
| 11.8 || Properties of functions represented by real power series || 431 | |||
|- | |||
| 11.9 || The Taylor's series generated by a function || 434 | |||
|- | |||
| 11.10 || A sufficient condition for convergence of a Taylor's series || 435 | |||
|- | |||
| 11.11 || Power-series expansions for the exponential and trigonometric functions || 435 | |||
|- | |||
| <nowiki>*</nowiki>11.12 || Bernstein's theorem || 437 | |||
|- | |||
| 11.13 || Exercises || 438 | |||
|- | |||
| 11.14 || Power series and differential equations || 439 | |||
|- | |||
| 11.15 || The binomial series || 441 | |||
|- | |||
| 11.16 || Exercises || 443 | |||
|- | |||
! colspan="3" | 12. VECTOR ALGEBRA | |||
|- | |||
| 12.1 || Historical introduction || 445 | |||
|- | |||
| 12.2 || The vector space of n-tuples of real numbers || 446 | |||
|- | |||
| 12.3 || Geometric interpretation for <math>n \leq 3</math> || 448 | |||
|- | |||
| 12.4 || Exercises || 450 | |||
|- | |||
| 12.5 || The dot product || 451 | |||
|- | |||
| 12.6 || Length or norm of a vector|| 453 | |||
|- | |||
| 12.7 || Orthogonality of vectors || 455 | |||
|- | |||
| 12.8 || Exercises || 456 | |||
|- | |||
| 12.9 || Projections. Angle between vectors in n-space || 457 | |||
|- | |||
| 12.10 || The unit coordinate vectors || 458 | |||
|- | |||
| 12.11 || Exercises || 460 | |||
|- | |||
| 12.12 || The linear span of a finite set of vectors || 462 | |||
|- | |||
| 12.13 || Linear independence || 463 | |||
|- | |||
| 12.14 || Bases || 466 | |||
|- | |||
| 12.15 || Exercises || 467 | |||
|- | |||
| 12.16 || The vector space <math>V_N(C)</math> of n-tuples of complex numbers || 468 | |||
|- | |||
| 12.17 || Exercises || 470 | |||
|- | |||
! colspan="3" | 13. APPLICATIONS OF VECTOR ALGEBRA TO ANALYTIC GEOMETRY | |||
|- | |||
| 13.1 || Introduction || 471 | |||
|- | |||
| 13.2 || Lines in n-space || 472 | |||
|- | |||
| 13.3 || Some simple properties of straight lines || 473 | |||
|- | |||
| 13.4 || Lines and vector-valued functions || 474 | |||
|- | |||
| 13.5 || Exercises || 477 | |||
|- | |||
| 13.6 || Planes in Euclidean n-space || 478 | |||
|- | |||
| 13.7 || Planes and vector-valued functions || 481 | |||
|- | |||
| 13.8 || Exercises || 482 | |||
|- | |||
| 13.9 || The cross product || 483 | |||
|- | |||
| 13.10 || The cross product expressed as a determinant || 486 | |||
|- | |||
| 13.11 || Exercises || 487 | |||
|- | |||
| 13.12 || The scalar triple product || 488 | |||
|- | |||
| 13.13 || Cramer's rule for solving a system of three linear equations || 490 | |||
|- | |||
| 13.14 || Exercises || 491 | |||
|- | |||
| 13.15 || Normal vectors to planes || 493 | |||
|- | |||
| 13.16 || Linear Cartesian equations for planes || 494 | |||
|- | |||
| 13.17 || Exercises || 496 | |||
|- | |||
| 13.18 || The conic sections || 497 | |||
|- | |||
| 13.19 || Eccentricity of conic sections || 500 | |||
|- | |||
| 13.20 || Polar equations for conic sections || 501 | |||
|- | |||
| 13.21 || Exercises || 503 | |||
|- | |||
| 13.22 || Conic sections symmetric about the origin || 504 | |||
|- | |||
| 13.23 || Cartesian equations for the conic sections || 505 | |||
|- | |||
| 13.24 || Exercises || 508 | |||
|- | |||
| 13.25 || Miscellaneous exercises on conic sections || 509 | |||
|- | |||
! colspan="3" | 14. CALCULUS OF VECTOR-VALUED FUNCTIONS | |||
|- | |||
| 14.1 || Vector-valued functions of a real variable || 512 | |||
|- | |||
| 14.2 || Algebraic operations. Components || 512 | |||
|- | |||
| 14.3 || Limits, derivatives, and integrals || 513 | |||
|- | |||
| 14.4 || Exercises || 516 | |||
|- | |||
| 14.5 || Applications to curves. Tangency || 517 | |||
|- | |||
| 14.6 || Applications to curvilinear motion. Velocity, speed, and acceleration || 520 | |||
|- | |||
| 14.7 || Exercises || 524 | |||
|- | |||
| 14.8 || The unit tangent, the principal normal, and the osculating plane of a curve || 525 | |||
|- | |||
| 14.9 || Exercises || 528 | |||
|- | |||
| 14.10 || The definition of arc length || 529 | |||
|- | |||
| 14.11 || Additivity of arc length || 532 | |||
|- | |||
| 14.12 || The arc-length function || 533 | |||
|- | |||
| 14.13 || Exercises || 535 | |||
|- | |||
| 14.14 || Curvature of a curve || 536 | |||
|- | |||
| 14.15 || Exercises || 538 | |||
|- | |||
| 14.16 || Velocity and acceleration in polar coordinates || 540 | |||
|- | |||
| 14.17 || Plane motion with radial acceleration || 542 | |||
|- | |||
| 14.18 || Cylindrical coordinates || 543 | |||
|- | |||
| 14.19 || Exercises || 543 | |||
|- | |||
| 14.20 || Applications to planetary motion || 545 | |||
|- | |||
| 14.21 || Miscellaneous review exercises || 549 | |||
|- | |||
! colspan="3" | 15. LINEAR SPACES | |||
|- | |||
| 15.1 || Introduction || 551 | |||
|- | |||
| 15.2 || The definition of a linear space || 551 | |||
|- | |||
| 15.3 || Examples of linear spaces || 552 | |||
|- | |||
| 15.4 || Elementary consequences of the axioms || 554 | |||
|- | |||
| 15.5 || Exercises || 555 | |||
|- | |||
| 15.6 || Subspaces of a linear space || 556 | |||
|- | |||
| 15.7 || Dependent and independent sets in a linear space || 557 | |||
|- | |||
| 15.8 || Bases and dimension || 559 | |||
|- | |||
| 15.9 || Exercises || 560 | |||
|- | |||
| 15.10 || Inner products, Euclidean spaces, norms || 561 | |||
|- | |||
| 15.11 || Orthogonality in a Euclidean space || 564 | |||
|- | |||
| 15.12 || Exercises || 566 | |||
|- | |||
| 15.13 || Construction of orthogonal sets. The Gram-Schmidt process || 568 | |||
|- | |||
| 15.14 || Orthogonal complements. Projections || 572 | |||
|- | |||
| 15.15 || Best approximation of elements in a Euclidean space by elements in a finite-dimensional subspace || 574 | |||
|- | |||
| 15.16 || Exercises || 576 | |||
|- | |||
! colspan="3" | 16. LINEAR TRANSFORMATIONS AND MATRICES | |||
|- | |||
| 16.1 || Linear transformations || 578 | |||
|- | |||
| 16.2 || Null space and range || 579 | |||
|- | |||
| 16.3 || Nullity and rank || 581 | |||
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| 16.4 || Exercises || 582 | |||
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| 16.5 || Algebraic operations on linear transformations || 583 | |||
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| 16.6 || Inverses || 585 | |||
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| 16.7 || One-to-one linear transformations || 587 | |||
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| 16.8 || Exercises || 589 | |||
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| 16.9 || Linear transformations with prescribed values || 590 | |||
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| 16.10 || Matrix representations of linear transformations || 591 | |||
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| 16.11 || Construction of a matrix representation in diagonal form || 594 | |||
|- | |||
| 16.12 || Exercises || 596 | |||
|- | |||
| 16.13 || Linear spaces of matrices || 597 | |||
|- | |||
| 16.14 || Isomorphism between linear transformations and matrices || 599 | |||
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| 16.15 || Multiplication of matrices || 600 | |||
|- | |||
| 16.16 || Exercises || 603 | |||
|- | |||
| 16.17 || Systems of linear equations || 605 | |||
|- | |||
| 16.18 || Computation techniques || 607 | |||
|- | |||
| 16.19 || Inverses of square matrices || 611 | |||
|- | |||
| 16.20 || Exercises || 613 | |||
|- | |||
| 16.21 || Miscellaneous exercises on matrices || 614 | |||
|- | |||
! colspan="2" | Answers to exercises || 617 | |||
|- | |||
! colspan="2" | Index || 657 | |||
|- | |- | ||
|} | |} | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
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