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MATHEMATICS

 
Basic Mathematic: Numbers, Fraction, Factors etc
 
 
Arithmetic
 
Basic Addition, Basic Subtraction, Division, Adding Decimals, Subtracting decimals, Dividing decimal, Greatest Common Divisor, Percent and decimals, ordering numeric expressions, Basic Multiplication, The Multiplication Tables, Lattice Multiplication
 
 
Order of operations, Adding/Subtracting negative numbers, Multiplying and dividing negative numbers, Adding and Subtracting fractions multiplying fractions, Dividing fractions, Level 1 Exponents, Level 2 Exponents, Level 3 Exponents, Negative Exponent Intuition, Exponent Rules Part, Simplifying radicals, Introduction to Logarithms, Unit Conversion, Speed translation, Introduction to logarithm properties, Scientific Notation
 
 
Linear Equations, Solving Inequalities, graphing lines, Graphing linear equations Slope and Y-intercept intuition, the Slope of a line, Equation of a line, Averages, Integer sums, Taking percentages, growing by a percentage, systems of Equations, Ratios
 
 
 Multiplying expressions, solving a quadratic by factoring, Complex Numbers, quadratic equation, using the quadratic ,Equation to solve 2nd degree polynomials, Completing the square, Quadratic Formula (proof) , Quadratic Formula Quadratic Inequalities, Introduction to functions, Functions, Domain of a function, Proof: log a + log b = log ab, Proof: A (log B) = log (B^A), log A - log B = log (A/B), Proofs of the logarithm properties: A (log B) = log (B^A) and log A - log B = log (A/B), Proof: log_a (B) = (log_x (B))/ (log_x (A)), Algebraic Long Division, Conic Sections, Circles, Ellipses, Hyperbolas, Identifying Conics
 
 
Foci of an Ellipse, Foci of a Hyperbola, Proof: Hyperbola Foci, Partial Fraction Expansion, Parabola Focus and Directrix, Two Passing Bicycles Word Problem, Passing Trains problem, Overtaking Word Problem
 
 
Matrices, Matrix multiplication ,Inverse Matrix, Matrices to solve a system of equations, Matrices to solve a vector combination problem ,Singular Matrices , 3-variable linear equations ,Introduction to Vectors, Parametric Representations of Lines, Linear Combinations and Span, Linear Independence ,Linear Subspaces
 
 
Vector Dot Product and Vector Length, Proving Vector Dot Product Properties ,Proof of the Cauchy-Schwarz Inequality, Proof of the Cauchy-Schwarz Inequality  Proof of the Cauchy-Schwarz Inequality, Vector Triangle Inequality, Vector Triangle Inequality  Proving the triangle inequality for vectors in Rn, Defining the angle between vectors, Defining the angle between vectors  Introducing the idea of an angle between two vectors, Defining a plane in R3 with a point and normal vector, Cross Product Introduction, Relationship between cross product and sin of angle, Dot and Cross Product Comparison/Intuition, Dot and Cross Product Comparison/Intuition  Dot and Cross Product Comparison/Intuition, matrices: Reduced Row Echelon Form , Matrix Vector Products, Null Space of a Matrix ,Calculating the null space of a matrix, Column Space of a Matrix ,Visualizing a Column Space as a Plane in R3,Proof: Any subspace basis has same number of elements, Dimension of the Null Space or Nullity , Dimension of the Column Space or Rank ,relation between basis cols and pivot cols  ,Showing that the candidate basis does span C(A)
 
 
Vector Transformations, Linear Transformations, Matrix Vector Products as Linear Transformations, Linear Transformations as Matrix Vector Products, Image of a subset under a transformation, im (T): Image of a Transformation, Preimage of a set, Preimage and Kernel Example, Sums and Scalar Multiples of Linear Transformations, Rotation in R3 around the X-axis, Unit Vectors, Matrix Product Examples, Matrix Product Associativity, Distributive Property of Matrix Products,
 
 
Introduction to the inverse of a function, Proof: Invertibility implies a unique solution to f(x)=y, Surjective (onto) and Injective (one-to-one) functions ,Relating invertibility to being onto and one-to-one ,Determining whether a transformation is onto , Exploring the solution set of Ax=b ,Matrix condition for one-to-one trans ,Simplifying conditions for invertibility  ,Showing that Inverses are Linear, Deriving a method for determining inverses,Finding Matrix Inverse ,Formula for 2x2 inverse , 3x3 Determinant , nxn Determinant ,Determinants along other rows/cols, Rule of Sarrus of Determinants ,Determinant when row multiplied by scalar, scalar multiplication of row ,Determinant when row is added, Determinant when row is added ,Duplicate Row Determinant, Determinant after row operations, Upper Triangular Determinant, Simpler 4x4 determinant ,Determinant and area of a parallelogram, Determinant as Scaling Factor ,Transpose of a Matrix ,Determinant of Transpose  Proof ,Transposes of sums and inverse, Transpose of a Vector
 
 
 
Rowspace and Left Nullspace ,Visualizations of Left Nullspace and Rowspace ,Orthogonal Complements, Rank(A) = Rank(transpose of A), dim(V) + dim(orthogonal complement of V)=n , Representing vectors in Rn using subspace members, Orthogonal Complement of the Orthogonal Complement, Orthogonal Complement of the Nullspace, Unique rowspace solution to Ax=b , Unique rowspace solution to Ax=b ,Rowspace Solution to Ax=b example ,Showing that A-transpose x A is invertibl, Projections onto Subspaces,Visualizing a projection onto a plane ,Visualizing a projection onto a plane  Visualizing a projection onto a plane., A Projection onto a Subspace is a Linear Transforma, Subspace Projection Matrix, Projection is closest vector in subspace, Least Squares Approximation , Coordinates with Respect to a Basis, Change of Basis Matrix, Invertible Change of Basis Matrix ,Transformation Matrix with Respect to a Basis ,Alternate Basis Transformation Matrix, Changing coordinate systems to help find a transformation matrix ,Introduction to Orthonormal Bases, Coordinates with respect to orthonormal bases , Projections onto subspaces with orthonormal bases, orthogonal matrices preserve angles and lengths
 
 
The Gram-Schmidt Process, Eigenvalues and Eigenvectors, Proof of formula for determining Eigenvalues ,Eigenvalues of a 3x3 matrix, Showing that an eigenbasis makes for good coordinate systems
 
 
Introduction to angles, Angles of parallel lines, similar triangles, Area and Perimeter, Circles: Radius, Diameter and Circumference, Area of a circle, Pythagorean Theorem, 45-45-90 Triangles, 30-60-90 Triangles
 
 
 
Basic Trigonometry, Radians and degrees, Using Trig Functions ,The unit circle definition of trigonometric function, Graph of the sine function, Graphs of trig functions, Determining the equation of a trigonometric function, Trigonometric Identities, Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b),Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b)
, Trig identities, Trigonometry word problems, Law of cosines, Proof: Law of Sines, Ferris Wheel Trig Problem, Polar Coordinates, and Inverse Trig Functions: Arcsin, Arctan, and Arccos
 
 
 
The Average , Sample vs. Population Mean ,Variance of a Population ,Sample Variance , Standard Deviation, Alternate Variance Formulas  ,Random Variables, Probability Density Functions, Binomial Distribution, Expected Value: E(X) ,Expected Value of Binomial Distribution  ,Poisson Process, Law of Large Numbers, Normal Distribution ,Standard Normal Distribution and the Empirical Rule
 
 
probability ,Permutations ,Combinations, Probability using Combinations, Conditional Probability and Combinations ,Random Variables ,Probability Density Functions, Binomial Distribution ,Law of Large Numbers 
 
 
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