Download Domain of sin(x) , cos(x) is R. Domain of tan(x) is R \ {(k + 2)π : k ∈ Z

Domain of sin(x) , cos(x) is R .
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Domain of tan(x) is R \ (k + 2 ) π : k ∈ Z .
Domain of cot(x) is R \ kπ : k ∈ Z .
Some examples
On the following pages, we show the graphs of several functions (polynomials and
trigonometric functions):
f (x) = 2x + 1 (blue), f (x) = −x + 1 (red)
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f (x) = sin(x)(blue), f (x) = cos(x) (red)
f (x) = tan(x) (blue), f (x) = cot(x) (red)
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f (x) = x4 − 2x3 − 2x2 + 4x + 2
Exponential function
exp(x) = ex , x ∈ R
(e the Euler number, e ≈ 2.71828)
Logarithm: ln(x) , x ∈ ]0 , ∞[ , the inverse function of the exponential function.
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Remark: Other basis a > 0, a 6= 1:
a
x
= exp x ln(a) ,
loga(x) =
ln(x)
,
ln(a)
x∈R
x ∈ ]0 , ∞[
Here are the graphs of the exponential function ex and ln(x):
f (x) = ex (blue), f (x) = ln(x) (red)
If f (x) is a function, the inverse g(x) of f (x) is a function such that g[f (x)] = x
for all x that are allowed for f . Not all functions have an inverse, for instance
there is no inverse for f (x) = x2.
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We obtain the equation for g(x) by solving f (x) = y for x, then we get an
expression g(y) = x, and then we simply replace x by y. This means that the
graph of the inverse fuinction g(x) can be obtained from the graph of f (x) by
reflecting it about the line with equation y = x.
This can be seen in the following picture for the exponential and logarithmic
function:
Inverse trigonometric functions
arcsin(x) , x ∈ [−1 , 1] ,
π π
has its values in − 2 , 2
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arccos(x) , x ∈ [−1 , 1] ,
arctan(x) , x ∈ R ,
arccot(x) , x ∈ R ,
has its values in [0 , π]
has its values in − π2 , π2
has its values in [0 , π]
f (x) = arcsin(x)(blue), f (x) = arccos(x) (red)
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f (x) = arctan(x)(blue), f (x) = arccot(x) (red)
1.3
The n-dimensional Euclidean space
Note that functions f : M → N have been defined for arbitrary sets M and N .
In most applications, M and N are not just real numbers but elements from the
Euclidean space.
Vectors: Addition, multiplication by scalars
For two vectors a = (a1 , a2, . . . , an ) und b = (b1, b2, . . . , bn), the vector a + b
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(the sum of a and b) is given by
a + b = ( a1 + b1, a2 + b2, . . . , an + bn) .
For a vector a = (a1, a2 , . . . , an) and a scalar (real number) λ, the vector λ a
or a λ is given by
λ a = a λ = ( λa1 , λa2 , . . . , λan) .
Remark:
−b = (−1)b = (−b1, −b2, . . . , −bn)
a − b = a + (−b) = ( a1 − b1, a2 − b2, . . . , an − bn)
a
= α1 a , (if α 6= 0)
α
Obvious properties
For vectors a, b, c in Rn and scalars λ, µ ∈ R :
(a + b) + c = a + (b + c) ,
λ (a + b) = λa + λb ,
a+b = b+a;
(λ + µ) a = λa + µa ,
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(subtraction)
λ (µ a) = (λ µ) a ;
a + 0 = a , a − a = 0 , null-vector 0 = (0, 0, . . . , 0) .
Sum and linear combinations of k vectors v1 , v2, . . ., vk of Rn :
k
X
vi = v1 + v2 + . . . + vk
i=1
k
X
λi vi = λ1v1 + λ2v2 + . . . + λkvk
(for scalars λ1 , . . . , λk ) .
i=1
A simple fact:
Every vector a = (a1, a2 , . . . , an) of Rn is a linear combination of the elementary
unit vectors e1, e2, . . . , en , where
(i = 1, 2, . . . , n)
ei = 0, . . . , 0, |{z}
1 , 0, . . . , 0 ,
i-th
since (obviously) :
a = a1 e1 + a2 e2 + . . . + an en
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Scalar product or inner product
For a = (a1 , . . . , an) and b = (b1, . . . , bn) :
n
X
a·b =
ai bi = a1b1 + a2 b2 + . . . + an bn
i=1
is called the scalar product or the inner product of the two vectors a und b.
Note: The result a · b is a scalar (real number), but not a vector.
Obvious properties:
a · b = b · a , (a + b) · c = a · c + b · c ,
(λ a) · b = λ (a · b) , (where λ ∈ R) ,
a · a ≥ 0 ; a · a = 0 only if a = 0 (the null-vector)
Length or norm of a vector
For a = (a1 , a2 , . . . , an ) :
|a| =
√
q
a·a =
a21 + a22 + . . . + a2n
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Note:
For two points (=vectors) of Rn ,
b = (b1, b2, . . . , bn) and c = (c1, c2, . . . , cn) ,
the distance of b and c is given by
p
|b − c| = (b1 − c1 )2 + (b2 − c2 )2 + . . . + (bn − cn )2
Properties:
|a| ≥ 0 ; |a| = 0 only if a = 0 ;
|λ a| = |λ| · |a| , (where λ ∈ R) ;
|b ± c| ≤ |b| + |c|
|b · c| ≤ |b| · |c|
(“triangle inequality”) ;
(“Cauchy-Schwarz inequality”)
The angle included by two vectors
For two vectors a = (a1, a2 , . . . , an ) and b = (b1, b2, . . . , bn)
(both not the null-vector) the angle included by a and b is given by
a·b ϕ = ∠(a, b) = arccos
.
Note: 0 ≤ ϕ ≤ π .
|a| · |b|
Equivalent formulation: a · b = |a| · |b| · cos(ϕ) .
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Orthogonality of two vectors
Two vectors a and b in Rn are said to be orthogonal
(or perpendicular), abbreviation: a ⊥ b , if a · b = 0 .
Note: a ⊥ b ⇐⇒ ∠(a, b) = π2 , (if a, b 6= 0).
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2
2.1
Vectors and matrices
Definitions
Let p and n be positive integers: p, n ∈ N = {1, 2, 3, . . .} . A (real) p × n
matrix is an arrangement of p · n real numbers in p rows and n columns. Each
real number is identified by the pair (i, j) representing the numbers of the row (i)
and column (j) in which it occurs, and is called the (i, j)-th entry of the matrix.


a11 a12 . . . a1n


 a21 a22 . . . a2n 
 ..
...
...
... 

 .
ap1 ap2 . . . apn
Here the (i, j)-th entry is aij ∈ R, (i = 1, . . . , p, j = 1, . . . , n).
Short notation: A = aij
i=1,...,p
j=1,...,n
or simply A.
Examples
1 −3 5
Special 2 × 3 matrix: A =
; special 3 × 2 matrix: B =
−2 0 4
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1 −3
2 0
1 1
!
.
For a p×n matrix A = aij
matrix B = bi,j i=1,...,q :
i=1,...,p
j=1,...,n
and a q ×m
j=1,...,m
A=B
⇐⇒
p = q, n = m, aij = bij for all i, j
Only one row : A 1 × n matrix is called a row vector (of dimension n),
A = a11 a12 . . . a1n , or a = (a1, a2, . . . , an) .
Only one column: A p × 1 matrix is called a column vector (of dimension p),
 


a1
a11
 


a 
a 
A =  ..21  , or a =  ..2  .
 . 
 . 
ap
ap1
We
distinguish between row vectors and column vectors. So, e.g., the vectors
3
and ( 3 , 5 ) are different.
5
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We write R
.
n instead of Rn×1
a11 a12 . . . a1n
 a21 a22 . . . a2n 
Let A =  .. .. .. .. 
.
.
.
.
ap1 ap2 . . . apn
The row vectors of the matrix A : are
i
a = ai1, ai2, . . . , ain , (the i-th row of A ) , for i = 1, . . . , p ,
The column
 vectors
 of the matrix A :
a1j
 a2j 
aj =  ..  ,
.
apj
So one may write:
Transposition
(the j-th column of A ) , for j = 1, . . . , n .


a1
 a2 
A =  ..  =
.
ap
a1, a2, . . . , an
For a row vector (of matrix size 1 × n, or of dimension n) , a = a1, a2, . . . , an
its transpose is:
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

a1
 a2 
at =  ..  (which is a column vector of size n × 1 ) .
.
an


b1
 b2 
For a column vector (of matrix size p × 1, or of dimension p), b =  ..  its
.
bp
transpose is:
bt = b1, b2, . . . , bp (which is a row vector of matrix size 1 × p (dimension p) .
For a p × n matrix A with rows a1, a2, . . . , ap :
The transpose At has the columns
(a1)t, (a2)t, . . . , (ap)t ;
so At is an n × p matrix.
In other words: The (i, j)-th entry of At equals
the (j, i)-th entry of A , for all i = 1, . . . , n ,
j = 1, . . . , p .
Examples: Some square matrices
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B=
0 −1
3 1
t
, B =
!
9 8 7
6 5 4
3 2 1
C=
D=
0 −1
−1 1
0 3
−1 1
, Ct =
= Dt ,
.
9 6 3
8 5 2
7 4 1
!
.
9 8 7
8 5 4
7 4 1
E=
!
Symmetric matrices
A matrix A is called symmetric, if At = A .
In particular: A symmetric matrix must be a
square matrix, i.e.,
of size n × n (for some n ∈ N).
2.2
Matrix Algebra
Addition, multiplication by scalars
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= Et .