McWeeny, R., Symmetry: An Introduction to Group Theory and Its Applications, Unabridged edition, Dover Publications, New York, 2002.. Poole, D., Student Solutions Manual for Poole’s Line
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Lay, D C., Linear Algebra and Its Applications, 3rd Edition, Addison Wesley, Boston, 2002.
Lial, M L., Student’s Solutions Manual for College Algebra, 3rd Sol Mn Edition, Addison Wesley, Boston,
2004.
Lial, M L., Hornsby, J., and Schneider, D I., College Algebra, 9th Edition, Addison Wesley, Boston, 2004 Lipschutz, S., 3,000 Solved Problems in Linear Algebra, McGraw-Hill, New York, 1989.
Lipschutz, S and Lipson, M., Schaum’s Outline of Linear Algebra, 3rd Edition, McGraw-Hill, New York,
2000.
MacDuffee, C C., The Theory of Matrices, Phoenix Edition, Dover Publications, New York, 2004.
McWeeny, R., Symmetry: An Introduction to Group Theory and Its Applications, Unabridged edition, Dover
Publications, New York, 2002.
Meyer, C D., Matrix Analysis and Applied Linear Algebra, Package Edition, Society for Industrial & Applied
Mathematics, University City Science Center, Philadelphia, 2001.
Mikhalev, A V and Pilz, G., The Concise Handbook of Algebra, Kluwer Academic, Dordrecht, Boston, 2002 Perlis, S., Theory of Matrices, Dover Ed Edition, Dover Publications, New York, 1991.
Poole, D., Linear Algebra: A Modern Introduction, Brooks Cole, Stamford, 2002.
Poole, D., Student Solutions Manual for Poole’s Linear Algebra: A Modern Introduction, 2nd Edition, Brooks
Cole, Stamford, 2005.
Rose, J S., A Course on Group Theory (Dover Books on Advanced Mathematics), Dover Publications, New
York, 1994.
Schneider, H and Barker, G Ph., Matrices and Linear Algebra, 2nd Edition (Dover Books on Advanced
Mathematics), Dover Publications, New York, 1989.
Scott, W R., Group Theory, Dover Publications, New York, 1987.
Shilov, G E., Linear Algebra, Rev English Ed Edition, Dover Publications, New York, 1977.
Strang, G., Introduction to Linear Algebra, 3rd Edition, Wellesley Cambridge Pr., Wellesley, 2003.
Strang, G., Linear Algebra and Its Applications, 4th Edition, Brooks Cole, Stamford, 2005.
Sullivan, M., College Algebra, 7th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2004.
Tobey, J and Slater, J., Beginning Algebra, 6th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2004 Tobey, J and Slater, J., Intermediate Algebra, 5th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2005 Trefethen, L N and Bau, D., Numerical Linear Algebra, Society for Industrial & Applied Mathematics,
University City Science Center, Philadelphia, 1997.
Turnbull, H W and Aitken, A C., An Introduction to the Theory of Canonical Matrices, Phoenix Edition,
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Vygodskii, M Ya., Mathematical Handbook: Higher Mathematics, Mir Publishers, Moscow, 1971.
Zassenhaus, H J., The Theory of Groups, 2nd Ed Edition, Dover Publications, New York, 1999.
Zhang, F., Matrix Theory, Springer, New York, 1999.
Trang 2Limits and Derivatives
6.1 Basic Concepts of Mathematical Analysis
6.1.1 Number Sets Functions of Real Variable
6.1.1-1 Real axis, intervals, and segments
The real axis is a straight line with a point O chosen as the origin, a positive direction, and
a scale unit
There is a one-to-one correspondence between the set of all real numbersR and the set
of all points of the real axis, with each real x being represented by a point on the real axis separated from O by the distance|x|and lying to the right of O for x >0, or to the left of O
for x <0
One often has to deal with the following number sets (sets of real numbers or sets on the real axis)
1 Sets of the form (a, b), (– ∞, b), (a, +∞), and (–∞, +∞) consisting, respectively, of
all x R such that a < x < b, x < b, x > a, and x is arbitrary are called open intervals
(sometimes simply intervals).
2 Sets of the form [a, b] consisting of all xR such that a≤x≤b are called closed intervals or segments.
3 Sets of the form (a, b], [a, b), (– ∞, b], [a, +∞) consisting of all x such that a < x≤b,
a≤x < b, x≤b , x≥a are called half-open intervals.
A neighborhood of a point x ◦ R is defined as any open interval (a, b) containing x ◦
(a < x ◦ < b) A neighborhood of the “point” + ∞, –∞, or ∞ is defined, respectively, as
any set of the form (b, + ∞), (–∞, c) or (–∞, –a) ∪ (a, +∞) (here, a≥ 0)
6.1.1-2 Lower and upper bound of a set on a straight line
The upper bound of a set of real numbers is the least number that bounds the set from above The lower bound of a set of real numbers is the largest number that bounds the set from
below
In more details: let a set of real numbers XR be given A number β is called its upper
bound and denoted sup X if for any xX the inequality x≤β holds and for any β1 < β there exists an x1 X such that x1> β1 A number α is called the lower bound of X and denoted inf X if for any xX the inequality x≥α holds and for any α1> α there exists
an x1X such that x1 < α1
Example 1 For a set X consisting of two numbers a and b (a < b), we have
inf X = a, sup X = b.
Example 2 For intervals (open, closed, and half-open), we have
inf(a, b) = inf[a, b] = inf(a, b] = inf[a, b) = a, sup(a, b) = sup[a, b] = sup(a, b] = sup[a, b) = b.
235
Trang 3236 LIMITS ANDDERIVATIVES
One can see that the upper and lower bounds may belong to a given set (e.g., for closed intervals) and may not (e.g., for open intervals).
The symbol +∞ (resp., –∞) is called the upper (resp., lower) bound of a set unbounded
from above (resp., from below)
6.1.1-3 Real-valued functions of real variable Methods of defining a function
1◦ Let D and E be two sets of real numbers Suppose that there is a relation between
the points of D and E such that to each xD there corresponds some y E, denoted by
y = f (x) In this case, one speaks of a function f defined on the set D and taking its values
in the set E The set D is called the domain of the function f , and the subset of E consisting
of all elements f (x) is called the range of the function f This functional relation is often denoted by y = f (x), f : D → E, f : x → y.
The following terms are also used: x is the independent variable or the argument; y is the dependent variable.
2◦ The most common and convenient way to define a function is the analytic method: the
function is defined explicitly by means of a formula (or several formulas) depending on the
argument x; for instance, y =2sin x +1
Implicit definition of a function consists of using an equation of the form F (x, y) =0,
from which one calculates the value y for any fixed value of the argument x.
Parametric definition of a function consists of defining the values of the independent variable x and the dependent variable y by a pair of formulas depending on an auxiliary variable t (parameter): x = p(t), y = q(t).
Quite often functions are defined in terms of convergent series or by means of tables or graphs There are some other methods of defining functions
3◦ The graph of a function is the representation of a function y = f (x) as a line on the plane
with orthogonal coordinates x, y, the points of the line having the coordinates x, y = f (x), where x is an arbitrary point from the domain of the function.
6.1.1-4 Single-valued, periodic, odd and even functions
1◦ A function is single-valued if each value of its argument corresponds to a unique value
of the function A function is multi-valued if there is at least one value of its argument
corresponding to two or more values of the function In what follows, we consider only single-valued functions, unless indicated otherwise
2◦ A function f (x) is called periodic with period T (or T -periodic) if f (x + T ) = f (x) for
any x.
3◦ A function f (x) is called even if it satisfies the condition f (x) = f (–x) for any x A
function f (x) is called odd if it satisfies the condition f (x) = –f (–x) for any x.
6.1.1-5 Decreasing, increasing, monotone, and bounded functions
1◦ A function f (x) is called increasing or strictly increasing (resp., nondecreasing) on a set
D ⊂ R if for any x1, x2D such that x1> x2, we have f (x1) > f (x2) (resp., f (x1)≥f (x2))
A function f (x) is called decreasing or strictly decreasing (resp., nonincreasing) on a set
D if for all x1, x2D such that x1> x2, we have f (x1) < f (x2) (resp., f (x1)≤f (x2) ) All
such functions are called monotone functions Strictly increasing or decreasing functions are called strictly monotone.
Trang 42◦ A function f (x) is called bounded on a set D if|f (x)| < M for all xD , where M is
a finite constant A function f (x) is called bounded from above (bounded from below) on a set D if f (x) < M (M < f (x)) for all xD , where M is a real constant.
6.1.1-6 Composite and inverse functions
1◦ Consider a function u = u(x), x
D , with values uE , and let y = f (u) be a function defined on E Then the function y = f u (x)
, xD , is called a composite function or the superposition of the functions f and u.
2◦ Consider a function y = f (x) that maps x
D into y E The inverse function of
y = f (x) is a function x = g(y) defined on E and such that x = g(f (x)) for all xD The
inverse function is often denoted by g = f– 1.
For strictly monotone functions f (x), the inverse function always exists In order to construct the inverse function g(y), one should use the relation y = f (x) to express x through y The function g(y) is monotonically increasing or decreasing together with f (x).
6.1.2 Limit of a Sequence
6.1.2-1 Some definitions
Suppose that there is a correspondence between each positive integer n and some (real or complex) number denoted, for instance, by x n In this case, one says that a numerical sequence (or, simply, a sequence) x1, x2, , x n , is defined Such a sequence is often
denoted by{x n}; xn is called the generic term of the sequence.
Example 1 For the sequence{n2– 2}, we have x1 = – 1, x2 = 2, x3 = 7, x4 = 14 , etc.
A sequence is called bounded (bounded from above, bounded from below) if there is a constant M such that|x n|< M (respectively, x n < M , x n > M ) for all n =1, 2,
6.1.2-2 Limit of a sequence
A number b is called the limit of a sequence x1, x2, , x n , if for any ε >0 there is
N = N (ε) such that|x n – b| < ε for all n > N
If b is the limit of the sequence{x n}, one writes lim
n→∞ x n = b or x n → b as n → ∞.
The limit of a constant sequence{x n = c} exists and is equal to c, i.e., lim
n→∞ = c In this
case, the inequality|x n – c| < ε takes the form0< ε and holds for all n.
Example 2 Let us show that lim
n→∞
n
n+ 1 =1.
Consider the difference n
n+ 1 –1 = 1
n+ 1 The inequality
1
n+ 1 < ε holds for all n >
1
ε – 1= N (ε) Therefore, for any positive ε there is N = 1
ε – 1such that for n > N we have n
n+ 1–1 < ε.
It may happen that a sequence {x n} has no limit at all, for instance, the sequence {x n}={(–1)n} A sequence that has a finite limit is called convergent
THEOREM(BOLZANO–CAUCHY) A sequence x nhas a finite limit if and only if for any
ε>0, there is N such that the inequality
|x n – x m|< ε holds for all n > N and m > N
Trang 5238 LIMITS ANDDERIVATIVES
6.1.2-3 Properties of convergent sequences
1 Any convergent sequence can have only one limit
2 Any convergent sequence is bounded From any bounded sequence one can extract
a convergent subsequence.*
3 If a sequence converges to b, then any of its subsequence also converges to b.
4 If{x n},{y n}are two convergent sequences, then the sequences{x n y n},{x n⋅y n}, and{x n /y n}(in this ratio, it is assumed that y n ≠ 0and lim
n→∞ y n ≠ 0) are also convergent and
lim
n→∞ (x n y n) = limn→∞ x n n→∞lim y n;
lim
n→∞ (cx n ) = c lim n→∞ x n (c = const);
lim
n→∞ (x n⋅y n) = limn→∞ x n⋅ lim
n→∞ y n;
lim
n→∞
x n
y n =
lim
n→∞ x n
lim
n→∞ y n
5 If{x n},{y n}are convergent sequences and the inequality x n≤y n holds for all n,
then lim
n→∞ x n≤ lim
n→∞ y n.
6 If the inequalities x n ≤ y n ≤ z n hold for all n and lim n→∞ x n = limn→∞ z n = b, then
lim
n→∞ y n = b.
6.1.2-4 Increasing, decreasing, and monotone sequences
A sequence {x n} is called increasing or strictly increasing (resp., nondecreasing) if the inequality x n+1> x n (resp., x n+1≥x n ) holds for all n A sequence{x n}is called decreasing
or strictly decreasing (resp., nonincreasing) if the inequality x n+1< x n (resp., x n+1≤x n)
holds for all n All such sequences are called monotone sequences Strictly increasing or decreasing sequences are called strictly monotone.
THEOREM Any monotone bounded sequence has a finite limit
Example 3 It can be shown that the sequence
1 + 1
n
n4
is bounded and increasing Therefore, it is
convergent Its limit is denoted by the letter e:
e= lim
n→∞
1 + 1
n
n
(e≈ 2 71828 ).
Logarithms with the base e are called natural or Napierian, and log e x is denoted by
ln x.
6.1.2-5 Properties of positive sequences
1◦ If a sequence x n (x n>0) has a limit (finite or infinite), then the sequence
y n= √ n
x1⋅x2 x n
has the same limit
* Let {x n} be a given sequence and let {n k}be a strictly increasing sequence with k and n kbeing natural numbers The sequence {x n }is called a subsequence of the sequence{x n}
Trang 62◦ From property1◦ for the sequence
x1, x x2
1,
x3
x2, ,
x n
x n–1,
x n+1
x n , ,
we obtain a useful corollary
lim
n→∞ n
√
x n= limn→∞ x x n+1
n ,
under the assumption that the second limit exists
Example 4 Let us show that lim
n→∞
n
n
√
n! = e.
Taking x n= n
n
n! and using property2◦, we get
lim
n→∞
n
n
√
n! = limn→∞
x n+1
x n = lim
n→∞
1 + 1
n
n
= e.
6.1.2-6 Infinitely small and infinitely large quantities
A sequence x n converging to zero as n → ∞ is called infinitely small or infinitesimal.
A sequence x n whose terms infinitely grow in absolute values with the growth of n
is called infinitely large or “tending to infinity.” In this case, the following notation is
used: lim
n→∞ x n= ∞ If, in addition, all terms of the sequence starting from some number
are positive (negative), then one says that the sequence x n converges to “plus (minus) infinity,” and one writes lim
n→∞ x n= +∞ lim
n→∞ x n= –∞ For instance, lim
n→∞(–1)n n2=∞,
lim
n→∞
√
n= +∞, lim
n→∞ (–n) = – ∞.
THEOREM(STOLZ) Let x n and y n be two infinitely large sequences, y n → +∞, and y n
increases with the growth of n (at least for sufficiently large n): y n+1> y n Then
lim
n→∞
x n
y n = limn→∞
x n – x n–1
y n – y n–1, provided that the right limit exists (finite or infinite)
Example 5 Let us find the limit of the sequence
z n= 1k+ 2k+· · · + n k
n k+1 .
Taking x n= 1k+ 2k+· · · + n k and y n = n k+1in the Stolz theorem, we get
lim
n→∞ z n= lim
n→∞
n k
n k+1– (n –1 )k+1.
Since (n –1 )k+1= n k+1– (k +1)n k+· · · , we have n k+1– (n –1 )k+1= (k +1)n k+· · · , and therefore
lim
n→∞ z n= lim
n→∞
n k
(k +1)n k+· · · =
1
k+ 1.
Trang 7240 LIMITS ANDDERIVATIVES
6.1.2-7 Upper and lower limits of a sequence
The limit (finite or infinite) of a subsequence of a given sequence x nis called a partial limit
of x n In the set of all partial limits of any sequence of real numbers, there always exists the largest and the least (finite or infinite) The largest (resp., least) partial limit of a sequence
is called its upper (resp., lower) limit The upper and lower limits of a sequence x n are denoted, respectively,
lim
n→∞ x n, n→∞lim x n.
Example 6 The upper and lower limits of the sequence x n= (– 1 )nare, respectively,
lim
n→∞ x n= 1 , lim
n→∞ x n= – 1
A sequence x nhas a limit (finite or infinite) if and only if its upper limit coincides with its lower limit:
lim
n→∞ x n= limn→∞ x n= limn→∞ x n.
6.1.3 Limit of a Function Asymptotes
6.1.3-1 Definition of the limit of a function One-sided limits
1◦ One says that b is the limit of a function f (x) as x tends to a if for any ε >0there is
δ = δ(ε) >0such that|f (x) – b| < ε for all x such that0<|x – a| < δ.
Notation: lim
x→a f (x) = b or f (x) → b as x → a.
One says that b is the limit of a function f (x) as x tends to + ∞ if for any ε >0there is
N = N (ε) >0such that|f (x) – b| < ε for all x > N
Notation: lim
x→+∞ f (x) = b or f (x) → b as x → +∞.
In a similar way, one defines the limits for x → –∞ or x → ∞.
THEOREM(BOLZANO–CAUCHY1) A function f (x) has a finite limit as x tends to a (a is assumed finite) if and only if for any ε >0there is δ >0such that the inequality
holds for all x1, x2such that|x1– a| < δand|x2– a| < δ.
THEOREM(BOLZANO–CAUCHY2) A function f (x) has a finite limit as x tends to + ∞
if and only if for any ε >0there isΔ > 0 such that the inequality (6.1.3.1) holds for all
x1>Δ and x2>Δ
2◦ One says that b is the left-hand limit (resp., right-hand limit) of a function f (x) as x
tends to a if for any ε >0there is δ = δ(ε) >0such that|f (x) – b| < ε for a – δ < x < a (resp., for a < x < a + δ).
Notation: lim
x→a–0f (x) = b or f (a –0) = b (resp., lim
x→a+0f (x) = b or f (a +0) = b)
6.1.3-2 Properties of limits
Let a be a number or any of the symbols ∞, +∞, –∞.
1 If a function has a limit at some point, this limit is unique
2 If c is a constant function of x, then lim
x→a c = c.
... fixed value of the argument x.Parametric definition of a function consists of defining the values of the independent variable x and the dependent variable y by a pair of formulas...
of x n In the set of all partial limits of any sequence of real numbers, there always exists the largest and the least (finite or infinite) The largest (resp., least) partial... data-page="7">
240 LIMITS ANDDERIVATIVES
6.1.2-7 Upper and lower limits of a sequence
The limit (finite or infinite) of a subsequence of a given sequence x nis called a partial limit