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Download Domain of sin(x) , cos(x) is R. Domain of tan(x) is R \ {(k + 2)π : k ∈ Z
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Domain of sin(x) , cos(x) is R . 1 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) 9 f (x) = sin(x)(blue), f (x) = cos(x) (red) f (x) = tan(x) (blue), f (x) = cot(x) (red) 10 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. 11 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. 12 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 13 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) 14 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 15 (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 , 16 (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 17 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 18 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) . 19 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). 20 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 21 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 22 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: 23 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 24 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 25 = Et .