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
Maths
