December 2, 2012
Math 10
Algebra.
Matrix-vector notation for systems of linear equations.
A 2-, 3-, …, n– dimensional vector can be written in the form of either a
row, or a column of 2, 3, …, n numbers. A 2x2, 3x3, …, nxn square
matrix is a table with (2, 3, …, n) columns and (2, 3, …, n) rows.
The multiplication of a matrix and a vector results in another vector,
according to the following rules:
,
,
and so on. Therefore, a system of linear equations can be written as an
equality between the two vectors (two vectors are equal if and only if
all components are equal). For example, the system of equations,
can be written in matrix-vector form as,
.
A system of linear equations where all constants on the right side are
zero,
,
is called homogeneous . Or, in the matrix-vector form,
.
Solving a linear system.
There are several algorithms for solving a system of linear equations.
The simplest method for solving a system of linear equations is to
repeatedly eliminate variables:
1. In the first equation, solve for one of the variables in terms of
the others.
2. Plug this expression into the remaining equations. This yields a
system of equations with one fewer equation and one fewer
unknown.
3. Continue until you have reduced the system to a single linear
equation.
4. Solve this equation, and then back-substitute until the entire
solution is found.
Let us first apply this method to a homogeneous system of 2 linear
equations:
We found that system has a unique solution, x = y = 0, provided that
expression
. Otherwise, it has an infinite number of
solutions, as the two equations are proportional to each other and
describe the same line. The expression
is called
determinant of the corresponding 2x2 matrix,
.
Let us now apply this method to a homogeneous system of 3 linear
equations shown above. This system has obvious solution, x = y = z = 0.
However, under some circumstances, it can have infinitely many
solutions. Let us see it.
,
If both coefficients in either of the first two equations are equal to
zero, it becomes an identity and the system has infinitely many
solutions. Using now the result obtained above for the homogeneous
system of two equations, we also see that the system has an infinite
number of solutions, if
.
This can be recast in a more conventional way of the determinant,
.
Solving a linear system by Gaussian elimination.
Another simple method of solving the system of linear equations is by
Gaussian elimination. In this method, one replaces equations with linear
combination of equations, such that one of the variables disappears.
Consider the example from the homework.
Homework review.
–
–
–
Solution of a linear system by Cramers rule.
Consider first a 2x2 system of linear equations,
It can be easily solved by the elimination of variables to obtain,
This can be written by using the determinants as defined above,
Similar formulas hold for systems with more than two variables, giving
the explicit solution for a system of equations in the form of the
quotients of certain determinants, known as the Cramers rule. Thus,
for the system of 3 equations considered above, we have
Sequence
In mathematics, a sequence is an ordered list of objects. Like a set, it
contains members (also called elements, or terms). The number of
ordered elements (possibly infinite) is called the length of the
sequence. Unlike a set, order matters, and exactly the same elements
can appear multiple times at different positions in the sequence.
For example, (M, A, R, Y) is a sequence of letters that differs from (A,
R, M, Y), as the ordering matters, and (1, 1, 2, 3, 5, 8), which contains
the number 1 at two different positions, is a valid sequence. Sequences
can be finite, as in this example, or infinite, such as the sequence of all
even positive integers (2, 4, 6,...).
Examples and notation
There are various and quite different notions of sequences in
mathematics, some of which are not covered by the notations
introduced below.
,
,
,
…
Note: Sometimes sequences start with an index of n = 0, so the
first term is actually a0. Then the second term would be a1.
Examples:
{1, 2, 3, 4 ,...} is a very simple sequence (and it is an infinite sequence)
{20, 25, 30, 35, ...} is also an infinite sequence
{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite
sequence)
{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles
{a, b, c, d, e} is the sequence of the first 5 letters alphabetically
{f, r, e, d} is the sequence of letters in the name "fred"
{0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they
are in order, it is an alternating order in this case)
is a sequence given by the formula, few first members of
this sequence are (1, 3, 6, 10, 15, 21…)
A more formal definition of a finite sequence with terms in a set S is a
function from {1, 2, ..., n} to S for some n > 0. An infinite sequence in S
is a function from {1, 2, ... } to S. For example, the sequence of prime
numbers (2,3,5,7,11, … ) is the function 1→2, 2→3, 3→5, 4→7, 5→11,
….
How are the following sequences called: (i)
, (ii)
?
Problem. Write down the first few terms of each of the following
sequences.
a.
b.
c.
where
The formula defining
digit of .
is called the formula of the general term of
the sequence.
For any sequence
formal sum,
the associated series is defined as the ordered
The sequence of partial sums
associated to a series
defined for each k as the sum of the sequence
from
to
is
,
Recap: cumulative reference material.
Solving a linear system
There are several algorithms for solving a system of linear equations.
The simplest method for solving a system of linear equations is to
repeatedly eliminate variables. This method can be described as
follows:
1. In the first equation, solve for one of the variables in terms of
the others.
2. Plug this expression into the remaining equations. This yields a
system of equations with one fewer equation and one fewer
unknown.
3. Continue until you have reduced the system to a single linear
equation.
4. Solve this equation, and then back-substitute until the entire
solution is found.
For example, consider the following system:
Solving the first equation for x gives x = 5 + 2z − 3y, and plugging this
into the second and third equation yields
Solving the first of these equations for y yields y = 2 + 3z, and plugging
this into the second equation yields z = 2. We now have:
Substituting z = 2 into the second equation gives y = 8, and substituting
z = 2 and y = 8 into the first equation yields x = −15. Therefore, the
solution set is the single point (x, y, z) = (−15, 8, 2).
Row reduction (Gaussian elimination)
Suppose the goal is to find and describe the solution(s), if any, of the
following system of linear equations:
The algorithm is as follows: eliminate x from all equations below , and
then eliminate y from all equations below . This will put the system
into triangular form. Then, using back-substitution, each unknown can
be solved for.
In the example, x is eliminated from by adding
eliminated from by adding to . Formally:
to
The result is:
Now y is eliminated from
by adding
to
:
. x is then
The result is:
This result is a system of linear equations in triangular form, and so
the first part of the algorithm is complete.
The last part, back-substitution, consists of solving for the knowns in
reverse order. It can thus be seen that
Then,
can be substituted into
, which can then be solved to obtain
Next, z and y can be substituted into
The system is solved.
Problem: solve the system
, which can be solved to obtain
Systems of equations. Geometrical view.
Consider a system of equations,
y
How many solutions can it have? Figure on
1
the right illustrates several possible cases.
x
1
What could be said of the coefficients a, b,
c, d, e, f?
What types of systems do figures below
illustrate?
(a)
(b)
y
1
0 1
(d)
1
0 1
x
y
(e)
1
x
1
(c)
y
y
1
0 1
x
(f)
y
1
0 1
x
x
Systems of equations
A system of equations is a set of two or more equations with the same
variables. A solution to a system of equations is a set of values for the
variables that satisfy all the equations simultaneously. In order to
solve a system of equations, one must find all the sets of values of the
variables that constitute solutions of the system.
For example, the solution of the system of two equations
is an infinite number of pairs
. For the system
solutions are two pairs of numbers,
System of linear equations (or linear system) is a collection of linear
equations involving the same set of variables. For example,
is a system of three equations in the three variables x, y, z. A solution
to a linear system is an assignment of numbers to the variables such
that all the equations are simultaneously satisfied. A solution to the
system above is given by
since it makes all three equations valid.
A general system of m linear equations with n unknowns can be written
as
Here
of the system,
and
are the unknowns,
are the coefficients
are the constant terms.
A solution of a linear system is an assignment of values to the variables
x1, x2, ..., xn such that each of the equations is satisfied. The set of all
possible solutions is called the solution set.
A linear system may behave in any one of three possible ways:
1. The system has infinitely many solutions.
2. The system has a single unique solution.
3. The system has no solution.
Geometric interpretation
For a system involving two variables (x and y), each linear equation
determines a line on the xy-plane. Because a solution to a linear system
must satisfy all of the equations, the solution set is the intersection of
these lines, and is hence either a line, a single point, or the empty set.
For three variables, each linear equation determines a plane in threedimensional space, and the solution set is the intersection of these
planes. Thus the solution set may be a plane, a line, a single point, or
the empty set.
The solution set for the
equations x − y = −1 and 3x +
General behavior
y = 9 is the single point
(2, 3).
A linear system in three
The solution set for two
variables determines a
equations in three variables
collection of planes. The
is usually a line.
intersection point is the
In general, the behaviorsolution.
of a linear system is determined by the
relationship between the number of equations and the number of
unknowns:
1. Usually, a system with fewer equations than unknowns has
infinitely many solutions. Such a system is also known as an
underdetermined system.
2. Usually, a system with the same number of equations and
unknowns has a single unique solution.
3. Usually, a system with more equations than unknowns has no
solution. Such a system is also known as an overdetermined
system.
In the first case, the dimension of the solution set is usually equal to n
− m, where n is the number of variables and m is the number of
equations.
The following pictures illustrate this trichotomy in the case of two
variables:
One equation
Two equations
Three equations
The first system has infinitely many solutions, namely all of the points
on the blue line. The second system has a single unique solution, namely
the intersection of the two lines. The third system has no solutions,
since the three lines share no common point.
Keep in mind that the pictures above show only the most common case.
It is possible for a system of two equations and two unknowns to have
no solution (if the two lines are parallel), or for a system of three
equations and two unknowns to be solvable (if the three lines intersect
at a single point). In general, a system of linear equations may behave
differently than expected if the equations are linearly dependent, or if
two or more of the equations are inconsistent.
Independence
The equations of a linear system are independent if none of the
equations can be derived algebraically from the others. When the
equations are independent, each equation contains new information
about the variables, and removing any of the equations increases the
size of the solution set. For linear equations, logical independence is
the same as linear independence.
For example, the equations
are not independent — they are the same equation when scaled by a
factor of two, and they would produce identical graphs. This is an
example of equivalence in a system of linear equations.
For a more complicated example, the equations
are not independent, because the third equation is
the sum of the other two. Indeed, any one of these
equations can be derived from the other two, and
any one of the equations can be removed without affecting the solution
set. The graphs of these equations are three lines that intersect at a
single point.
Consistency
The equations 3x + 2y = 6 and 3x + 2y = 12 are
inconsistent.
A linear system is consistent if it has a solution,
and inconsistent otherwise. When the system is
inconsistent, it is possible to derive a
contradiction from the equations, that may
always be rewritten such as the statement 0 = 1.
For example, the equations
are inconsistent. In fact, by subtracting the first equation from the
second one and multiplying both sides of the result by 1/6, we get 0 = 1.
The graphs of these equations on the xy-plane are a pair of parallel
lines.
It is possible for three linear equations to be inconsistent, even though
any two of them are consistent together. For example, the equations
are inconsistent. Adding the first two equations together gives 3x + 2y
= 2, which can be subtracted from the third equation to yield 0 = 1.
Note that any two of these equations have a common solution. The
same phenomenon can occur for any number of equations.
In general, inconsistencies occur if the left-hand sides of the
equations in a system are linearly dependent, and the constant terms
do not satisfy the dependence relation. A system of equations whose
left-hand sides are linearly independent is always consistent.
Equivalence
Two linear systems using the same set of variables are equivalent if
each of the equations in the second system can be derived algebraically
from the equations in the first system, and vice-versa. Two systems
are equivalent if either both are inconsistent or each equation of any
of them is a linear combination of the equations of the other one. It
follows that two linear systems are equivalent if and only if they have
the same solution set.
Equations and inequalities with logarithms and exponents.
Equations where variable x is an argument of a composite function,
, where g(x) is some other function – linear, or a polynomial,
- can often be solved by a substitution,
, where y is a new
variable.
Example 1.
.
Denote
. Then the equation is equivalent to
, which has two solutions,
and
. Our
substitution implies
, so only the second case is possible solution,
leading to
. Both sides being positive, this equation is
equivalent to

, which is
a regular quadratic equation.
.
Both the exponential,
, and the logarithmic,
,
functions are monotonic within their respective domains of definition.
For a > 1 both functions are monotonously increasing, while they are
monotonously decreasing for a < 1. Hence, the following identities hold,
Example 2.

,
,

,
,

,
,
.

,
,
.



Transformation of functions


Horizontal translation: g(x)=f(x+c) . The graph is translated c units
to the left if c 0 and c units to the right if c 0 .
Vertical translation: g(x)=f(x)+k . The graph is translated k units
upward if k 0 and k units downward if k 0.

Change of scale: g(x)=f(ax) . The graph is "compressed" if a

and "stretched out" if a 1. In addition, if a 0 the graph is
reflected about the y -axis.
Change of amplitude: g(x)=Af(x) . The amplitude of the graph is
increased by a factor of A if A

1
1 and decreased by a factor of A
if A 1. In addition, if A 0 the graph is inverted.
Reflection over the x-axis: -f (x) reflects f (x) over the x-axis.
Algebra 9 recap: Logarithm
The logarithm of a number is the exponent
by which another fixed value, the base, has
to be raised to produce that number. For
example, the logarithm of 1000 to base 10 is
3, because 1000 is 10 to the power 3: 1000 =
103 = 10 × 10 × 10. More generally, if x = by,
then y is the logarithm of x to base b, and is
written logb(x), so log10(1000) = 3.
Logarithms were introduced by John Napier in the early 17th century
as a means to simplify calculations. They were rapidly adopted by
scientists, engineers, and others to perform computations more easily
and rapidly, using slide rules and logarithm tables. These devices rely
on the following facts—important in their own right:
Product, quotient, power, and root
The logarithm of a product is the sum of the logarithms of the
numbers being multiplied; the logarithm of the ratio of two numbers is
the difference of the logarithms. Therefore, the logarithm of the p-th
power of a number is p times the logarithm of the number itself; the
logarithm of a p-th root is the logarithm of the number divided by p.
The following table lists these identities with examples:
Formula
product
quotient
power
Example
root
Change of base
The logarithm logb(x) can be computed from the logarithms of x and b
with respect to an arbitrary base k using the following formula:
Fractional powers. Exponential function.
For any integer
and natural
,
defines powers for rational values of exponent. It is readily proven
using the multiplication and the addition of integer powers and roots
that same rules apply in this case,
(cf homework).
In order to define the exponential function we need to extend the
definition of powers to all real numbers, i. e. to irrational numbers. This
can be done using the following “asymptotic” reasoning.
Intervals of monotonic behavior. For a > 1 the value of ap increases
when p increases. For 0 < a < 1 the value of ap decreases when p
increases. For rational p = m/n this can be straightforwardly proven by
finding the common denominator of p = m/n < q = r/s (case of negative p
should be considered - cf. Gelfand&Shen, Algebra, p. 126).
Consequently, we can extend the definition of powers to irrational
numbers x, such as
, as follows.
Definition. For an irrational
, and
that for any rational q less than x,
number greater that x,
Similarly, for
,
is a number such that
, while for any rational
,
,
It is important to mention that in order to make this definition correct
we must prove that such a number exists and is unique. This is done in
calculus.
Now, using the above definition we have a way to calculate, say,
, to
any given accuracy. In order to do so, we must simply find a rational
number p that is close enough to
and compute
. In order to
improve the accuracy, we may choose another number, q, yet closer to
, and use it for the computation, and so on.
We can obtain a sequence of rational numbers approaching
(and
for any rational p) by using the continuous fraction representation,
What are the coefficients a, b, and c here?
Exponential function is a function of the form f(x) = bx for a fixed
base b, which could be any positive real number. Exponential functions
are characterized by the fact that their rate of growth is proportional
to their value. For example, suppose we start with a bank deposit such
that its growth rate at any time is proportional to the amount of money
in it. After t years the amount of money in the bank deposit will then
be m0at (an exponential function) where m0 is the initial deposit and
some a > 1 is the interest (growth) rate.
The shape of the graph of y = bx depends on whether b < 1, b = 1, or b >
1 as shown on the right. The green graph is the graph of bx (b > 1),
while the red and the blue graphs show the linear and the cubic
functions, correspondingly.
What are the domain and range of exponential
functions?
The graph illustrates how exponential growth
(green) surpasses both linear (red) and cubic
(blue) growth.
Exponential growth
growth
Linear growth
Cubic