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EM (Mathematics), Vector - Coggle Diagram

- - - - A matrix is a rectangular array of numbers or functions which we will enclose in brackets
    - - The numbers (or functions) are called entries or, less commonly, elements of the matrix
    - - Rows are the horizontal lines of entries
    - - Columns are the vertical lines of entries
    - - Matrices having just a single row or column
      - Row Vector
      - Column Vector
      - Entries are called components
      - We shall denote vectors by lowercase boldface letters a, b or by its general component in brackets, a = [aj]
    - - mxn matrix (read m by n matrix) we mean a matrix with m rows and n columns
      - A = [ajk ]
    - - If m = n, we call A an n x n square matrix
      - Main Diagonal
    - - A matrix of any size m x n
        is called a rectangular matrix
    - - Addition
        
        Element-Wise Addition
        
        Properties
        
        Associative
        
        Commutative
      - Matrix Multiplication
        
        [m x n] [n x p] = [m x p]
        
        Not Commutative AB != BA generally
        
        A.I = I.A = A
        
        A.A^-1 = A^-1.A = I
        
        A.0 = 0.A = 0
        
        Associative A(BC) = (AB)C
        
        AB = 0 and A != 0 and B != 0, then both A and B are singular
        
        AB = AC => B = C if A^-1 exists
      - Transpose
        
        Properties
        
        (AB)T = BTAT
        
        (A+B)T = AT + BT
        
        (AT)^-1 = (A^-1)T
    - - Symmetric Matrices
        
        Symmetric Matrices
        
        AT = A
        
        Symmetric matrices are square matrices whose transpose equals the matrix itself
        
        The eigenvalues of a real symmetric matrix are real
        
        The upper triangle and the lower triangle are mirror images of each other
        
        Skew-Symmetric Matrices
        
        Skew-symmetric matrices are square matrices whose transpose equals minus the matrix
        
        A T = -A
        
        Each eigenvalue of the real skew-symmetric matrix A is either 0 or a purely imaginary number
        
        The rank of a skew-symmetric matrix is an even number
        
        Any square matrix can be expressed as the sum of symmetric and skew symmetric matrix
        
        (A + AT)/2 + (A - AT)/2
      - Triangular Matrices
        
        Upper triangular matrices
        
        Upper triangular matrices are square matrices that can have nonzero entries only on and above the main diagonal, whereas any entry below the diagonal must be zero
        
        Lower Triangular Matrices
        
        Lower triangular matrices can have nonzero entries only on and below the main diagonal, whereas any entry below the diagonal must be zero
        
        Any entry on the main diagonal of a triangular matrix may be zero or not
      - Diagonal Matrices
        
        These are square matrices that can have nonzero entries only on
        the main diagonal
        
        Any entry above or below the main diagonal must be zero
        
        Scalar Matrix
        
        If all the diagonal entries of a diagonal matrix S are equal, say, c, we call S a scalar matrix because multiplication of any square matrix A of the same size by S has the same effect as the multiplication by a scalar
        
        AS = SA = cA
        
        Identity Matrix
        
        A scalar matrix, whose entries on the main diagonal are all 1, is called a unit matrix (or identity matrix) and is denoted by In or simply by I
        
        AI = IA = A
      - Hermitian Matrices
        
        Hermitian
        
        A = (Aconjugate)T
        
        aij = ajiconjugate
        
        Diagonal elements are real
        
        Eigen Values are real
        
        Skew Hermitian
        
        A = -(A')T
        
        aij = -ajiconjugate
        
        Diagonal elements are 0
        
        The eigenvalues of a skew-Hermitian matrix are all purely imaginary (and possibly zero)
        
        Any square matrix can be expressed as the sum of hermitian and skew hermitian matrix Link
      - Orthogonal Matrices
        
        A^-1 = AT
        
        Product of a matrix and its transpose gives Identity matrix
      - Singular
        
        Singular
        
        |A| = 0
        
        Non-Singular
        
        |A| != 0
        
        The sum of two singular n × n matrices may be non-singular
        
        The sum of two n × n non-singular matrices may be singular
      - Unitary
        
        A.(Aconjugate)T = I
        
        A^-1 = (Aconjugate)T
        
        Involuntary Matrix
        
        A is its inverse
        
        Idempotent Matrix
        
        A.A = A
        
        Nilpotent Matrix
        
        A^k = 0 class k
        
        Normal Matrix
        
        A complex square matrix A is normal if it commutes with its conjugate transpose A*
        
        A*(Aconjugate)T = (Aconjugate)T*A
      - Definite Symmetric Matrix
        
        A symmetric nxn real matrix M is said to be positive-definite if the scalar zTMz is strictly positive for every non-zero column vector z of n real numbers
    - - Linear Systems of Equations
        
        Gauss Elimination/Back Substitution
        
        Gauss Elimination
        
        Elementary Row Operations
        
        Interchange of two rows
        
        Addition of a constant multiple of one row to another row
        
        Multiplication of a row by a nonzero constant c
        
        Produces Row-Equivalent System
        
        Row-Equivalent Systems
        
        Row-equivalent linear systems have the same set of solutions.
        
        Determinism
        
        Determined
        
        Underdetermined
        
        An underdetermined system of linear equations has an infinite number of solutions
        
        Fewer equations than unknowns
        
        If there is linear dependency of rows then the reduced set of equations will be underdetermined
        
        Just providing number of equations is not enough, reduce to row echelon form to decide the rank
        
        Overdetermined
        
        Consistency
        
        consistent
        
        Unique or infinite solutions
        
        Inconsistent
        
        No solutions
        
        Rank of Coefficient Matrix != Rank of Augmented Matrix
        
        Trick
        
        Reduce coefficient matrix to row echelon form and apply the same to the augmented column and if the ranks are not same for augmented and coefficient matrix then it is inconsistent
        
        If rank of coefficient matrix is n then the values in augmented row does not matter as rank of augmented matrix will be n regardless
        
        Non-Homogeneous
        
        Types of Systems
        
        3 more items...
        
        B != 0
        
        Homogeneous
        
        B = 0
        
        2 more items...
        
        Homogeneous systems are always consistent as they always have trivial solution or infinite solutions
        
        Row Echelon Form
        
        Rows of zeros, if present, are the last rows
        
        In each nonzero row, the leftmost nonzero entry is farther
        to the right than in the previous row
        
        [A | b] = [R | f]
        
        Ax = b and Rx = f have identical solution
        
        Determine Solutions
        
        No solution
        
        2 more items...
        
        Unique Solution
        
        2 more items...
        
        Infinitely many solutions
        
        4 more items...
        
        The leading coefficient can be any number. However, the majority of Linear Algebra textbooks do state that the leading coefficient must be the number 1 Link
        
        Reduced Row Echelon Form
        
        The first non-zero number in the first row (the leading entry) is the number 1
        
        For every subsequent row, the number 1 must be further to the right.
        
        The leading entry in each row must be the only non-zero number in its column
        
        Any non-zero rows are placed at the bottom of the matrix
        
        Back Subtitution
        
        A shorter term for systems of linear equations is just linear systems
        
        A linear system of m equations in n unknowns x1,x2,... ,xn is a set of equations of the form
        
        a11x1 + .... a1nxn = b1
        a21x1 + .... a2nxn = b2
        ....
        am1x1 + .... amnxn = bm
        
        a11,.. , amn are given numbers, called the coefficients
        of the system. b1,... , bm on the right are also given numbers
        
        If all the b j are zero, then it is called a homogeneous system
        
        If at least one b j is not zero, then it is called a nonhomogeneous system
        
        A solution of this is a set of numbers x1 ,... , x n that satisfies all the m equations
        
        A solution vector of (1) is a vector x whose components form a solution of (1)
        
        If the system is homogeneous, it always has at least the trivial solution x1 = 0,... , xn = 0
        
        Matrix Form of the Linear System
        
        From the definition of matrix multiplication we see that the m equations of (1) may be written as a single vector equation
        
        Ax = b
        
        A = Coefficient Matrix
        
        Augment Matrix
        
        Coefficient Matrix + b
        
        The augmented matrix determines the system completely
        
        LU Decomposition
        
        Every matrix can be expressed as the product of a lower and an upper triangular matrix, provided all the principal minors are
        non-singular
        
        AX = B => LUX = B
        
        LY = B where Y = UX
        
        Solve X using backward substitution in UX = Y
        
        Solve Y using forward substitution in LY = B
        
        The U and L matrices both have diagonal elements
        
        One of them may be all 1s (either U or L)
        
        Possibility of decomposition
        
        The kth leading submatrix of A is denoted Ak and is the k × k matrix found by looking only at the top k rows and leftmost k columns
        
        An invertible matrix A has an LU decomposition provided that all its leading submatrices have non-zero determinants
        
        Example
      - Linear Independence and Dependence of Vectors
        
        c1a(1) + c2a(2) + ... + cma(m) = 0
        
        If above equation holds only for all ci =0 then the vectors are linearly independent
        
        We can express at least one of the vectors as a linear combination of the other vectors
        
        a(1) = k2a(2) + ... + kma(m)
        where k j = -c j / c 1
        
        Some k j ’s may be zero. Or even all of them, namely, if a (1) = 0
        
        Rank of a Matrix
        
        The rank of a matrix A is the maximum number of linearly independent row vectors of A. It is denoted by rank A
        
        The rank of a matrix is invariant under elementary row operations.
        Row-Equivalent matrices have the same rank
        
        If determinant is not zero then rank = number of rows
        
        For square matrix we can calculate determinant to check if there are any dependent rows
        
        Check (n-1)x(n-1) sub-matrices
        
        if all 0s then rank is not n-1
        
        If at least one non-zero then rank is n-1
        
        Repeat with lower sized matrices
        
        For mxn matrix the rank is <= min(m,n)
        
        Rank(A, B) = min(Rank(A), Rank(B))
        
        Rank(A*AT) = Rank(AT*A) = Rank(A) = Rank(AT)
        
        Nullity of a Matrix
        
        Number of 0 rows in row echelon form
        
        Relations
        
        Rank(AB)≤min(Rank(A),Rank(B))
        
        Rank(A+B)≤Rank(A)+Rank(B)
        
        Inverse
        
        Rank(A) = Rank(A^-1) = n
        
        Only full rank matrices has inverse
        
        The rank of a matrix can be defined as being the number of non-zero eigenvalues of the matrix
        
        Tricks
        
        Directly check if any row can be expressed as the some of other rows
        
        Rank is given as 1 and we have to find the value of a variable for which the rank holds
        
        Check values for which a row becomes a multiple of some other row
        
        Check if other rows also follow the same factor
        
        Linear Independence and
        Dependence of Vectors
        
        Consider p vectors that each have n components
        
        these vectors are linearly independent if the matrix formed, with these vectors as row vectors, has rank p
        
        These vectors are linearly dependent if that matrix has rank less than p
        
        Rank in Terms of Column Vectors
        
        The rank r of a matrix A equals the maximum number of linearly independent column vectors of A
        
        Hence A and its transpose A T have the same rank
        
        Linear Dependence of Vectors
        
        Consider p vectors each having n components. If n p, then these vectors are linearly dependent
        
        Vector Space
        
        Consider a nonempty set V of vectors where each vector has the same number of components
        
        If, for any two vectors a and b in V, we have that all their linear combinations c1a + c2b (c1, c2 any real numbers) are also elements of V
        
        For any three vectors
        a,b,c in V
        
        a + b = b + a
        
        (a + b) + c = a + (b + c)
        
        a + 0 = 0
        
        a + (-a) = 0
        
        basis
        
        A linearly independent set in V consisting of a maximum possible number of vectors in V is called a basis for V
        
        A set of vectors is a basis for
        a vector space V if
        
        the vectors in the set are linearly independent
        
        any vector in V can be expressed as a linear combination of the vectors in the set
        
        span
        
        The set of all linear combinations of given vectors a(1) ,... , a(p) with the same number of components is called the span of these vectors
        
        a span is a vector space
        
        If in addition, the given vectors a (1) ,..., a (p) are linearly independent, then they form a basis for that vector space
        
        Subspace
        
        a nonempty subset of V (including V itself) that forms a vector space with respect to the two algebraic operations (addition and scalar multiplication) defined for the vectors of V
        
        The vector space R n consisting of all vectors with n components (n real numbers) has dimension n
        
        Row/Column/NULL Space
        
        Row Space
        
        we call the span of the row vectors the row space of A
        
        Column Space
        
        the span of the column vectors of A is called the column space of A
        
        The row space and the column space of a matrix A have the same dimension, equal to rank A
        
        Null Space
        
        the solution set of the homogeneous system Ax = 0 is a
        vector space, called the null space of A, and its dimension is called the nullity of A
        
        Eigen Vector/Value
        
        Av = λv
        
        λ = Eigen Value
        
        Solving Eigen Value
        
        Av = λv
        
        Av − λIv = 0
        
        | A − λI | = 0
        
        Solve the determinant to find λ
        
        Involve solving quadratic or higher equations
        
        | A − λI | is the Characteristic Polynomial
        
        3x3 Trick
        
        2 more items...
        
        | A − λI | = 0 is Characteristic Equation
        
        Characteristic Equation is homogeneous has trivial/non-trivial solution
        
        Properties
        
        Sum of Eigen Values = Trace
        
        Product of Eigen Values = Determinant
        
        Eigen Values of A^2 = Square of Eigen Value of A
        
        Algebraic Multiplicity
        
        Number of times a particular eigen value repeats is its Algebraic Multiplicity
        
        Geometric Multiplicity
        
        Number of distinct Linearly Independent eigen vectors for a particular eigen value is its geometric multiplicity
        
        GM = n - r
        
        1 more item...
        
        Algebraic Multiplicity >= Geometric Multiplicity
        
        If A is an n×n real and symmetric matrix, then rank(A) =
        the total number of nonzero eigenvalues of A
        
        AT has the same eigen values as A
        
        Complex eigenvalues always appear as conjugate pairs
        
        Eigenvalues of A^n = nth power of eigen values, n can be -1 for inverse
        
        Eigenvalues of adj(A) = |A|/Eigenvalue
        
        Eigenvalues of A + kI = eigenvalue of A + k
        
        For UTM or LTM or diagonal matrix, the eigen values are the diagonal elements
        
        Eigenvalues of Idempotent matrix is 0, 1
        
        Eigenvalues of hermitian/symmetric matrix are real
        
        Eigenvalues of Skew Hermitian matrix are 0 or purely imaginary and for Skew-Symmetric its only 0
        
        Eigenvalues of involuntary matrix is 1, -1
        
        Eigenvalues of orthogonal matrix have modulus 1
        
        Cayley-Hamilton Theorem
        
        Every square matrix satisfies its characteristic equation
        
        3 more items...
        
        Tricks
        
        If sum of all rows/columns is same then that value is the eigen value
        
        Can be solved using substitution in a complex expression Link
        
        Cancellation Trick
        
        Av = λv will give some equations
        
        Add some of them can lead to cancellation of terms from LHS and RHS leaving λ behind
        
        The set of eigenvalues of a matrix is also called its spectrum
        
        Row transforms involve adding a multiple of one row to another row which changes the trace of the matrix (most of the time), but the trace is the sum of the eigenvalues, so (at least one of) the eigenvalues must change, as well.
        
        v = Eigen Vector
        
        Solve Eigen Vector
        
        Av = λv
        
        We have A and λ
        
        Eigenvector is any non-zero multiple of this solution
        
        v is a column vector x, y, z, ....
        
        Solutions
        
        LI Linearly Independent
        
        LD Linearly Dependent
        
        Properties
        
        A set of eigenvectors of A, each corresponding to a different eigenvalue of A, is a linearly independent set
        
        Eigenvectors are never 0
        
        If eigenvalue is real, then the corresponding eigenvectors are also real
        
        If eigenvalue is complex, then the corresponding eigenvectors are also complex
        
        A^m -> λ^m -> X
        
        Eigenvectors of real symmetric matrix are orthogonal vectors
        
        Example
        
        One eigen value can have multiple eigen vectors
        
        We have a single equation when evaluating the eigen vector components and we end with multiple vectors due to dependency
        
        The same eigenvalue can have multiple eigenvectors that are not only not equal but are also orthogonal
        
        One eigen vector can have single eigen value
        
        Unit Eigenvectors
        
        Magnitude of eigen vector should be 1
        
        For n dimensional vector x0^2 + x1^2 + ... + xn^2 = 1
        
        Av = λv
        
        vTAv = vTλv = λvTv = λ since vTv = 1 for unit vector
        
        Diagonalization
        
        P = Matrix of eigen vectors = Model Vector
        
        Let Vi(a column vector) be the eigen vector of λi
        
        ith column of P is Vi
        
        λi is also the present in the ith column of D
        
        D = Diagonal matrix of eigen values in the same order as eigen values in P
        
        Example for 3x3
        
        [λ0 0 0]
        [0 λ1 0]
        [0 0 λ2]
        
        A = PDP^-1
        
        e^A = Pe^AP^-1
        
        An nxn matrix A is diagonalizable iff and only if it has n linearly independent eigen vectors
        
        Usage
        
        A^n = PD^nP^-1
        
        exp(A) = Pe^DP^-1
        
        e^D is obtained by replacing the diagonal elements with e^λ in D
    - - Trace
        
        The trace of a square matrix A, denoted tr(A) is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of A
        
        Properties
        
        tr(A + B) = tr(A) + tr(B)
        
        tr(A) = tr(AT)
        
        A matrix and its transpose have the same trace
        
        tr(ABT) = tr(ATB) = tr(BAT) = tr(BTA)
        
        Cyclic Property
        
        tr(ABCD) = tr(BCDA) = tr(CDAB) = tr(DABC)
        
        Trace of a matrix product
        
        tr(AB) != tr(A)tr(B)
        
        tr(AB) = tr(BA)
        
        tr(cA) = c tr(A)
        
        Tr(AAT) is the sum of the squares of all entries of A
        
        Tr(AAT) = 0, then A = 0.
    - - Adjoint
        
        Transpose of the cofactor matrix
        
        2x2 Trick
        
        Swap diagonal and change sign of non-diagonal elements
      - Inverse
        
        Adj(A)/det(A)
        
        A should not be singular for det(A) to exist
        
        For orthogonal matrix A^-1 = AT
        
        Inverse of I is I
        
        Inverse of diagonal matrix : Replace by 1/x for each element
        
        (AB)^-1 = B^-1A^-1
        
        (A^-1)^-1 = adj(A^-1)/det(A^-1)
        
        Special Case
        
        Orthogonal Matrix
        
        A^-1 = AT
        
        Diagonal Matrix
        
        Reciprocal of diagonal elements
        
        Unitary Matrix
        
        A^-1 = (Aconjugate)T
        
        Trick
        
        For inverse, if MCQ is given try each option instead of calculating inverse
        
        Calculate value for single element to check quickly
      - Determinant
        
        Cofactor
        
        Aij=(−1)^(i+j)det(Mij)
        
        Cofactor Matrix
        
        A matrix with elements that are the cofactors, term-by-term, of a given square matrix
        
        Minor
        
        Mij = (n – 1) x (n – 1) matrix obtained by deleting the ith row and jth column
        
        Algorithm
        
        Properties
        
        Determinant of a matrix with two identical rows or columns is 0
        
        Determinant of a matrix with at least 1 row or column = 0 is 0
        
        It one row/column is a multiple of another then the determinant is 0
        
        Properties
        
        |A| = |AT|
        
        |A^-1| = 1 / |A|
        
        |AB| = |A||B|
        
        B = kA => |B| = k^n|A|
        
        On interchange of row/columns n times B = (-1)^n A
        
        Take care of this point while comparing matrices through row/col transform
        
        Determinant of UTM, LTM, Diagonal Matrix = Product of Diagonal Elements
        
        det(A^-1) = 1/det(A)
        
        det(AT) = det(A)
        
        Tricks
        
        Pick row with maximum number of zeros to reduce computation
        
        Tricks
        
        Scaling row
        
        If a row is scaled by a constant factor, we can multiply the determinant by its inverse to preserve the value
        
        If sum all rows leads to a constant row then we can scale to make it all 1s to make it easier to calculate
        
        Make as many zeros as possible, then it becomes much simpler to calculate the determinant
        
        Standard Formulas
        
        Form 1
        
        1 a a2 a3
        1 b b2 b3
        1 c c2 c3
        1 d d2 d3
        
        (a-b)(a-c)(a-d)(b-c)(b-d)(c-d)
    - - a*x1^2 + 2*b*x*y + c*y^2
      - [x y] * [a b] * [x]
        ..........[b c] * [y]
    - - A block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices
      - Block diagonal matrices
        
        A block diagonal matrix is a block matrix that is a square matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrices
      - Properties
  - - - Monotonic Functions
        
        Increasing
        
        f′(x) ≥ 0 ∀ x ∈ (a, b)
        
        In order for the function y = f(x) to be increasing on the interval
        (a, b), it is necessary and sufficient that the first derivative of the function be non-negative everywhere in this interval
        
        Decreasing
        
        In order for the function y = f(x) to be increasing on the interval
        (a, b), it is necessary and sufficient that the first derivative of the function be non-positive everywhere in this interval
        
        f′(x) ≤ 0 ∀ x ∈ (a, b)
      - Limits
        
        Indeterminate Forms
        
        Quotient
        
        ∞ / ∞
        
        0 / 0
        
        Power
        
        1^∞
        
        ∞^0
        
        0^0
        
        Subtraction
        
        ∞ - ∞
        
        Product
        
        0 x ∞
        
        L Hospital Rule
        
        Limits of f(x) / g(x) with form 0 / 0 and ∞ / ∞ can be evaluated with f'(x) / g'(x)
        
        Conversion to L Form
        
        0 x ∞ => ∞ / ∞
        
        ∞ / ∞ forms can be simplified by dividing numerator and denominator n (where n -> ∞) (f(x)/n) / (g(x)/n)
        
        lim can be moved across functions
        
        Definition
        
        Limit
        
        Let I be an open interval containing c , and let f be a function defined on I , except possibly at c . The limit of f(x) , as x approaches c , is L , denoted by limx→cf(x)=L means that given any ϵ>0 , there exists δ>0 such that for all x≠c , if |x−c|<δ , then |f(x)−L|<ϵ
        
        y -tolerance ϵ is given first and then the limit will exist if we can find an x-tolerance δ that works
        
        Left-Hand Limit
        
        Let I be an open interval containing c, and let f be a function defined on I, except possibly at c. The limit of f(x), as x approaches c from the left, is L, or, the left--hand limit of f at c is L, denoted by limx→c−f(x)=L
        
        Right-Hand Limit
        
        Let I be an open interval containing c , and let f be a function defined on I , except possibly at c . The limit of f(x) , as x approaches c from the right, is L , or, the right--hand limit of f at c is L , denoted by limx→c+f(x)=L
        
        Let f be a function defined on an open interval I containing c . Then limx→cf(x)=L if, and only if, limx→c−f(x)=Landlimx→c+f(x)=L.
        
        Piece-Wise Defined Function
        
        Standard Limits
        
        Limit 1
        
        Techniques
        
        Use log on the expression and then use exponentiation on the result. The log can be moved inside the limit as log is a continuous function.
      - Continuity
        
        Continuity at a point
        
        Conditions
        
        lim x→a f(x) exists
        
        lim x→a f(x)=f(a)
        
        f(a) is defined
        
        A function is discontinuous at a point a if it fails to be continuous at a
        
        Polynomials and rational functions are continuous at every point in their domains
        
        Discontinuity
        
        Types
        
        Removal Discontinuities
        
        f has a removable discontinuity at a if limx→af(x) exists.
        
        Jump Discontinuities
        
        f has a jump discontinuity at a if limx→a−f(x) and limx→a+f(x) both exist, but limx→a−f(x)≠limx→a+f(x)
        
        Infinite Discontinuities
        
        f has an infinite discontinuity at a if limx→a−f(x)=±∞ or limx→a+f(x)=±∞
      - Differentiability
        
        Derivative
        
        The derivative of a real valued function f(x) wrt x is the function f'(x) and is defined as lim h->0 f(x+h)-f(x)/h
        
        Standard Derivatives
        
        x^x
        
        A function is said to be differentiable if the derivative of the function exists at all points of its domain
      - Problems
      - Intermediate Value Theorem
        
        In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval
        
        Finding roots of a function in an interval
        
        Bolzano's Theorem
        
        Suppose that f(x) is a continuous function on a closed interval [a, b] and takes the values of the opposite sign at the extremes, and there is at least one c (a, b) such that f(c) = 0
      - Mean Value Theorem
        
        Cacuchy Theorem
        
        f and g are continuous in [a, b]
        
        f and g are differentiable in (a, b)
        
        ∃ c in (a, b) where (f(b) - f(a))/(g(b) - g(a)) = f'(c) / g'(c)
        
        Rolle's Theorem
        
        f(a) = f(b)
        
        f is continuous in [a, b]
        
        f is differentiable in (a, b)
        
        ∃ c in (a, b) where f'(c) = 0
        
        Lagrange Theorem
        
        f(a) != f(b)
        
        f is continuous in [a, b]
        
        f is differentiable in (a, b)
        
        ∃ c in (a, b) where f'(c) = (f(b) - f(a))/(b - a)
        
        Technique
        
        Define new function using specific offsets like f(y) and f(y+1) based on the conditions and then use the property of the new function to create a new relation betwen f(y) and f(y+1) etc.
      - Taylor Series
        
        Definition
        
        What is nth term?
        
        Series starts from 0, so power will be (x-a)^(n-1)
        
        McLaurin Series
        
        Taylor Series expanded around 0
        
        Common Series
        
        Set 1
      - Maxima/Minima
        
        f'(x) is 0 for maxima/minima
        
        The derivative changes from -ve to +ve for minima
        
        The derivative changes from +ve to -ve for maxima
        
        f''(x) sign is +ve for minima and -ve for maxima
        
        Critical
        
        Critical Point
        
        A critical point of a function of a single real variable, f(x), is a value x0 in the domain of f where it is not differentiable or its derivative is 0 (f ′(x0) = 0)
        
        Critical Value
        
        A critical value is the image under f of a critical point
        
        Stationary Point
        
        A stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero
        
        Inflexion Point
        
        An Inflection Point is where a curve changes from Concave upward to Concave downward (or vice versa) Link
        
        A point x=a is called point of inflection if f ''(a)=0 and f '''(a) !=0 Link
        
        Find extrema
        
        Polynomial
        
        For each 0 of first derivative find the value of second derivative
        
        If second derivative is 0 then it is not extrema
        
        Find 0s of first derivative
        
        Piecewise Function
        
        Take derivative
        
        Check the increase/decreasing behavior in each piece
      - Integration
        
        Definite
        
        Limits are specified
        
        Integration is area under the curve
        
        Substitution Technique
        
        Change Variable
        
        Change Limits
        
        Change dx
        
        Swap Limits
        
        Add -ve sign and swap limits
        
        Symmetric limit -a to a
        
        For odd function the integral will be 0
        
        2 time the integral from 0 to a
        
        Use trigonometric relation
        
        Wallis Theorem
        
        Only Sin or Cos
        
        Both Sin and Cos
        
        Property
        
        f(na - x) = f(x)
        
        Kings Formula
        
        Replace variable x with (lower limit + upper limit - x)
        
        Special Case
        
        Integration by parts
        
        Hindus Method
        
        Formula
        
        ILATE
        
        Partial Fractions
        
        Exponential in denominator trick
        
        Euler/Gamma Function
        
        Gamma
        
        Integer is factorial
        
        Indefinite
        
        Limits not specified
        
        Double Integrals
        
        Order of Integration
        
        Cases
        
        Independent case
        
        If functions are separable then integrate individually
        
        Dependent Case
        
        Limit of y is a function of x
        
        Limits Provided
        
        Limits have to be inferred
        
        Check strictly increasing/decreasing
        
        Check the value of the function at the two limits of the interval
        
        If the values are are required, then check if the derivative can become at 0 at any point
        
        If it cannot become 0, then it is strictly increasing
        
        If the function is inverse of the required based on the values at the limits, then it does not meet the criteria
  - - - Types
        
        Elementary Event
        
        Certain Event
        
        Impossible Event
        
        Mutually Exclusive Events
        
        Exhaustive Events
        
        Independent Events
        
        Equally Likely Events
        
        Complimenmtary Events
    - - Relative Frequency
      - Subjective
      - Classical
    - - P(B/A)
    - - Inclusion-Exclusion Principles
      - Complementary Probability
      - If B1, B2, Bn,..., are pairwise disjoint events of positive
        probability, then P(A) = P(A/B1)*P(B1) + ... + P(A/Bn)*P(Bn)
      - Conditional Probability Rule
        
        P(A/B) = P(A ∩ B) / P(B)
      - Bayes' Theorem
        
        P(Bi/A) = P(Bi)P(A/Bi)/SUM(P(Bi)P(A/Bi))
      - Rule of Total Probability
        
        P(E) = P(A)*P(E/A) + P(B)*P(E/B)
        
        A and B are mutually exclusive a exhaustive events
    - - Random Variable
        
        Any variable whose value is subject to variations due
        to randomness is called a random variable
        
        It can take a different set of values from a sample space
        in which each value has an associated probability
        
        Types
        
        Discrete Random Variable
        
        Discrete random variable is a variable that can take a
        value from a discrete range of values
        
        Properties
        
        SUM P(x)=1
        
        E(x) = SUM(xP(x))
        
        V(x) = E(x^2) - (E(x))^2
        
        Types
        
        Binomial Distribution
        
        Binomial Formula
        
        6 more items...
        
        The binomial distribution gives the discrete probability distribution Pp(n|N) of obtaining exactly n successes out of N Bernoulli trials (where the result of each Bernoulli trial is true with probability p and false with probability q=1-p) Link
        
        Poisson Distribution
        
        Mean&Variance
        
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        Problems
        
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        Properties
        
        3 more items...
        
        Probability
        
        2 more items...
        
        Exponential Distribution
        
        λe^(-λx) for x >= 0 and 0 for x < 0
        
        Mean
        
        1 more item...
        
        Variance
        
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        Cumulative Distribution
        
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        Uniform Distribution
        
        1/(b-a) for a<= x <= b and 0 elsewhere
        
        Mean
        
        1 more item...
        
        Variance
        
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        Cumulative Distribution
        
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        Continuous Random Variable
        
        Continuous random variable is a variable that can
        take a value from a continuous range of values
        
        Properties
        
        Cumulative Distribution Function
        
        E(x)
        
        V(x)
        
        P(a < x < b) = P(a <= x <= b) = P(a <= x < b) = P(a < x <= b)
        
        Types
        
        General Continuous Distribution
        
        Uniform Distribution
        
        Exponential Distribution
        
        Normal Distribution
  - - - Median
        
        Raw
        
        Even
        
        ((n/2+1)th value + n/2 th value) / 2
        
        Odd
        
        (n+1)/2 th value
        
        Grouped Frequency
        
        L + ((n/2 - cf)/f)*H
      - Mode
        
        Raw
        
        Number with highest frequency
        
        Grouped Frequency
        
        L + ((fm - f1) / (2fm - f1 -f2)) X H
      - Mean
        
        Raw
        
        SUM(Xi) / N
        
        Grouped Frequency
        
        SUM(Fi x Mi)/SUM(Fi)
        
        Discrete Frequency
        
        SUM(Fi x Xi)/SUM(Fi)
      - Relation
        
        Positively Skewed Distribution
        
        Mode <= Median <= Mean
        
        Negatively Skewed Distribution
        
        Mean <= Median <= Mode
        
        Symmetric Distribution
        
        Mean = Median = Mode
    - - Mean Deviation
        
        Discrete Frequency
        
        Grouped Frequency
        
        Raw Data
      - Standard Deviation
        
        Discrete Frequency
        
        Grouped Frequency
        
        Raw Data
      - Coefficient of Variation
        
        STD/Mean
        
        To compare two or more series which are measured in
        different units, we cannot use measures of dispersion.
        Thus, we require those measures which are independent
        of units