Online courses directory (107)

Extending our understanding of numbers below 0. Thinking about "absolute" distance from 0. Negative Numbers Introduction. Negative numbers on the number line. Number line 3. Ordering Negative Numbers. Ordering negative numbers. Number opposites. Number opposites. Negative Numbers Introduction. Number opposites. Example: Adding negative numbers. Example: Adding integers with different signs. Adding negative numbers. Adding and subtracting negative number examples. Why Subtracting a Negative Equivalent to Adding a Positive. Adding and subtracting negative numbers. Negative number word problems 1. Negative number word problems. Adding decimals of different signs word problem. Adding/Subtracting negative numbers. Multiplying Positive and Negative Numbers. Why a Negative Times a Negative is a Positive. Why a Negative Times a Negative Makes Intuitive Sense. Dividing Positive and Negative Numbers. Example: Multiplying numbers with different signs. Multiplying and dividing negative numbers. Multiplying and dividing negative numbers. Absolute Value and Number Lines. Absolute Value 1. Finding absolute values. Absolute Value of Integers. Comparing Absolute Values. Comparing absolute values. Absolute value word problems exercise. Absolute value word problems. Constructing and interpreting absolute value. Constructing and interpreting absolute value. Negative Numbers Introduction. Negative numbers on the number line. Number line 3. Ordering Negative Numbers. Ordering negative numbers. Number opposites. Number opposites. Negative Numbers Introduction. Number opposites. Example: Adding negative numbers. Example: Adding integers with different signs. Adding negative numbers. Adding and subtracting negative number examples. Why Subtracting a Negative Equivalent to Adding a Positive. Adding and subtracting negative numbers. Negative number word problems 1. Negative number word problems. Adding decimals of different signs word problem. Adding/Subtracting negative numbers. Multiplying Positive and Negative Numbers. Why a Negative Times a Negative is a Positive. Why a Negative Times a Negative Makes Intuitive Sense. Dividing Positive and Negative Numbers. Example: Multiplying numbers with different signs. Multiplying and dividing negative numbers. Multiplying and dividing negative numbers. Absolute Value and Number Lines. Absolute Value 1. Finding absolute values. Absolute Value of Integers. Comparing Absolute Values. Comparing absolute values. Absolute value word problems exercise. Absolute value word problems. Constructing and interpreting absolute value. Constructing and interpreting absolute value.

Understanding and solving exponents without algebra. Introduction to exponents. Raising a number to the 0th and 1st power. Powers of 1 and -1. Powers of fractions. Powers of zero. Exponent example 1. Exponent example 2. Positive and zero exponents. Understanding Square Roots. Square roots of perfect squares. Approximating Square Roots. Estimating square roots. Simplifying square roots. Simplifying radicals. Square Roots and Real Numbers. Simplifying radicals. Simplifying radicals 2. Adding and simplifying radicals. Adding and subtracting radicals. Finding Cube Roots. Cube root of a non-perfect cube. Simplifying a cube root. Cube roots. Cube roots 2. Patterns in zeros exercise. Patterns in zeros. Exponent Rules Part 1. Exponent Rules Part 2. Exponent Properties Involving Quotients. Exponent rules. Negative exponents. Negative Exponent Intuition. Zero, Negative, and Fractional Exponents. Negative exponents. Basic fractional exponents. Negative fractional exponent examples. Negative fractional exponent examples 2. Fractional exponents. Fractional exponents with numerators other than 1. Fractional exponents 2. Introduction to scientific notation. Scientific Notation. Scientific notation intuition. Scientific Notation Examples. Scientific Notation I. Scientific Notation Example 2. Scientific notation. Orders of magnitude exercise example 1. Orders of magnitude exercise example 2. Orders of magnitude. Multiplying in Scientific Notation. Multiplying in scientific notation example. Dividing in scientific notation example. Multiplying and dividing in scientific notation. Multiplying and dividing scientific notation. Simplifying a complicated expression into scientific notation. Calculating red blood cells in the body using scientific notation. Computing in scientific notation. Introduction to exponents. Raising a number to the 0th and 1st power. Powers of 1 and -1. Powers of fractions. Powers of zero. Exponent example 1. Exponent example 2. Positive and zero exponents. Understanding Square Roots. Square roots of perfect squares. Approximating Square Roots. Estimating square roots. Simplifying square roots. Simplifying radicals. Square Roots and Real Numbers. Simplifying radicals. Simplifying radicals 2. Adding and simplifying radicals. Adding and subtracting radicals. Finding Cube Roots. Cube root of a non-perfect cube. Simplifying a cube root. Cube roots. Cube roots 2. Patterns in zeros exercise. Patterns in zeros. Exponent Rules Part 1. Exponent Rules Part 2. Exponent Properties Involving Quotients. Exponent rules. Negative exponents. Negative Exponent Intuition. Zero, Negative, and Fractional Exponents. Negative exponents. Basic fractional exponents. Negative fractional exponent examples. Negative fractional exponent examples 2. Fractional exponents. Fractional exponents with numerators other than 1. Fractional exponents 2. Introduction to scientific notation. Scientific Notation. Scientific notation intuition. Scientific Notation Examples. Scientific Notation I. Scientific Notation Example 2. Scientific notation. Orders of magnitude exercise example 1. Orders of magnitude exercise example 2. Orders of magnitude. Multiplying in Scientific Notation. Multiplying in scientific notation example. Dividing in scientific notation example. Multiplying and dividing in scientific notation. Multiplying and dividing scientific notation. Simplifying a complicated expression into scientific notation. Calculating red blood cells in the body using scientific notation. Computing in scientific notation.

Introduction to Ratios (new HD version). Ratios as Fractions in Simplest Form. Simplifying Rates and Ratios. Expressing ratios as fractions. Describing ratios exercise. Describing ratios. Ratio word problem exercise example 1. Ratio word problem exercise example 2. Ratio word problems. Proportion validity example. Solving ratio problems with tables exercise. Solving ratio problems with tables exercise 2. Solving ratio problems with tables exercise 3. Solving ratio problems with tables. Usain Bolt's Average Speed. Finding Unit Rates. Finding Unit Prices. Rate problems 0.5. Multiple rates word problem. Rate problems 2. Converting pounds to ounces. Converting Gallons to quarts pints and cups. Converting Yards into Inches. Performing arithmetic calculations on units of volume. Application problems involving units of weight. Solving application problems involving units of volume. Gas cost for a roadtrip. Squirrel survival. Converting units of length. Unit Conversion Example: Drug Dosage. Unit Conversion with Fractions. Units. Speed translation. Unit conversion within the metric system. Converting within the metric system. Perimeter and Unit Conversion. Applying the Metric System. Conversion between metric units. U.S. Customary and Metric units. Converting Fahrenheit to Celsius. Comparing Celsius and Farenheit temperature scales. Introduction to Ratios (new HD version). Ratios as Fractions in Simplest Form. Simplifying Rates and Ratios. Expressing ratios as fractions. Describing ratios exercise. Describing ratios. Ratio word problem exercise example 1. Ratio word problem exercise example 2. Ratio word problems. Proportion validity example. Solving ratio problems with tables exercise. Solving ratio problems with tables exercise 2. Solving ratio problems with tables exercise 3. Solving ratio problems with tables. Usain Bolt's Average Speed. Finding Unit Rates. Finding Unit Prices. Rate problems 0.5. Multiple rates word problem. Rate problems 2. Converting pounds to ounces. Converting Gallons to quarts pints and cups. Converting Yards into Inches. Performing arithmetic calculations on units of volume. Application problems involving units of weight. Solving application problems involving units of volume. Gas cost for a roadtrip. Squirrel survival. Converting units of length. Unit Conversion Example: Drug Dosage. Unit Conversion with Fractions. Units. Speed translation. Unit conversion within the metric system. Converting within the metric system. Perimeter and Unit Conversion. Applying the Metric System. Conversion between metric units. U.S. Customary and Metric units. Converting Fahrenheit to Celsius. Comparing Celsius and Farenheit temperature scales.

If you can take one figure and flip, shift and rotate (not resize) it to be identical to another figure, then the two figures are congruent. This topic explores this foundational idea in geometry. Congruent Triangles and SSS. SSS to Show a Radius is Perpendicular to a Chord that it Bisects. Other Triangle Congruence Postulates. Two column proof showing segments are perpendicular. Finding Congruent Triangles. Congruency postulates. More on why SSA is not a postulate. Perpendicular Radius Bisects Chord. Congruent Triangle Proof Example. Congruent Triangle Example 2. Congruent triangles 1. Congruent triangles 2. Congruent legs and base angles of Isosceles Triangles. Equilateral Triangle Sides and Angles Congruent. Equilateral and Isosceles Example Problems. Triangle types. Triangle angles 1. Another Isosceles Example Problem. Example involving an isosceles triangle and parallel lines. Figuring out all the angles for congruent triangles example. Basic Triangle Proofs Module Example. Basic Triangle Proofs Module Example 2. Basic triangle proofs. Fill-in-the-blank triangle proofs example 1. Fill-in-the-blank triangle proofs example 2. Fill-in-the-blank triangle proofs. Wrong statements in triangle proofs example 1. Wrong statements in triangle proofs. Problem involving angle derived from square and circle. Congruent Triangles and SSS. SSS to Show a Radius is Perpendicular to a Chord that it Bisects. Other Triangle Congruence Postulates. Two column proof showing segments are perpendicular. Finding Congruent Triangles. Congruency postulates. More on why SSA is not a postulate. Perpendicular Radius Bisects Chord. Congruent Triangle Proof Example. Congruent Triangle Example 2. Congruent triangles 1. Congruent triangles 2. Congruent legs and base angles of Isosceles Triangles. Equilateral Triangle Sides and Angles Congruent. Equilateral and Isosceles Example Problems. Triangle types. Triangle angles 1. Another Isosceles Example Problem. Example involving an isosceles triangle and parallel lines. Figuring out all the angles for congruent triangles example. Basic Triangle Proofs Module Example. Basic Triangle Proofs Module Example 2. Basic triangle proofs. Fill-in-the-blank triangle proofs example 1. Fill-in-the-blank triangle proofs example 2. Fill-in-the-blank triangle proofs. Wrong statements in triangle proofs example 1. Wrong statements in triangle proofs. Problem involving angle derived from square and circle.

Similar Triangle Basics. Similarity Postulates. Similar triangles 1. Similar Triangle Example Problems. Similar triangles 2. Similarity Example Problems. Solving similar triangles 1. Similarity example where same side plays different roles. Challenging Similarity Problem. Finding Area Using Similarity and Congruence. Similar triangles. Similar triangles (part 2). Similar Triangle Basics. Similarity Postulates. Similar triangles 1. Similar Triangle Example Problems. Similar triangles 2. Similarity Example Problems. Solving similar triangles 1. Similarity example where same side plays different roles. Challenging Similarity Problem. Finding Area Using Similarity and Congruence. Similar triangles. Similar triangles (part 2).

Triangles are not always right (although they are never wrong), but when they are it opens up an exciting world of possibilities. Not only are right triangles cool in their own right (pun intended), they are the basis of very important ideas in analytic geometry (the distance between two points in space) and trigonometry. Pythagorean Theorem. The Pythagorean theorem intro. Pythagorean Theorem 1. Pythagorean Theorem 2. Pythagorean Theorem 3. Pythagorean theorem. Introduction to the Pythagorean Theorem. Pythagorean Theorem II. Garfield's proof of the Pythagorean Theorem. Bhaskara's proof of Pythagorean Theorem. Pythagorean Theorem Proof Using Similarity. Another Pythagorean Theorem Proof. 30-60-90 Triangle Side Ratios Proof. 45-45-90 Triangle Side Ratios. 30-60-90 Triangle Example Problem. Special right triangles. Area of a Regular Hexagon. 45-45-90 Triangles. Intro to 30-60-90 Triangles. 30-60-90 Triangles II. Pythagorean Theorem. The Pythagorean theorem intro. Pythagorean Theorem 1. Pythagorean Theorem 2. Pythagorean Theorem 3. Pythagorean theorem. Introduction to the Pythagorean Theorem. Pythagorean Theorem II. Garfield's proof of the Pythagorean Theorem. Bhaskara's proof of Pythagorean Theorem. Pythagorean Theorem Proof Using Similarity. Another Pythagorean Theorem Proof. 30-60-90 Triangle Side Ratios Proof. 45-45-90 Triangle Side Ratios. 30-60-90 Triangle Example Problem. Special right triangles. Area of a Regular Hexagon. 45-45-90 Triangles. Intro to 30-60-90 Triangles. 30-60-90 Triangles II.

You probably like triangles. You think they are useful. They show up a lot. What you'll see in this topic is that they are far more magical and mystical than you ever imagined!. Circumcenter of a Triangle. Circumcenter of a Right Triangle. Three Points Defining a Circle. Area Circumradius Formula Proof. 2003 AIME II Problem 7. Point Line Distance and Angle Bisectors. Incenter and incircles of a triangle. Inradius Perimeter and Area. Angle Bisector Theorem Proof. Angle Bisector Theorem Examples. Angle bisector theorem. Triangle Medians and Centroids. Triangle Medians and Centroids (2D Proof). Medians divide into smaller triangles of equal area. Exploring Medial Triangles. Proving that the Centroid is 2-3rds along the Median. Median Centroid Right Triangle Example. Proof - Triangle Altitudes are Concurrent (Orthocenter). Common Orthocenter and Centroid. Review of Triangle Properties. Euler Line. Euler's Line Proof. Constructing a perpendicular bisector using a compass and straightedge. Constructing a perpendicular line using a compass and straightedge. Constructing an angle bisector using a compass and straightedge. Compass Constructions. Circumcenter of a Triangle. Circumcenter of a Right Triangle. Three Points Defining a Circle. Area Circumradius Formula Proof. 2003 AIME II Problem 7. Point Line Distance and Angle Bisectors. Incenter and incircles of a triangle. Inradius Perimeter and Area. Angle Bisector Theorem Proof. Angle Bisector Theorem Examples. Angle bisector theorem. Triangle Medians and Centroids. Triangle Medians and Centroids (2D Proof). Medians divide into smaller triangles of equal area. Exploring Medial Triangles. Proving that the Centroid is 2-3rds along the Median. Median Centroid Right Triangle Example. Proof - Triangle Altitudes are Concurrent (Orthocenter). Common Orthocenter and Centroid. Review of Triangle Properties. Euler Line. Euler's Line Proof. Constructing a perpendicular bisector using a compass and straightedge. Constructing a perpendicular line using a compass and straightedge. Constructing an angle bisector using a compass and straightedge. Compass Constructions.

Not all things with four sides have to be squares or rectangles! We will now broaden our understanding of quadrilaterals!. Quadrilateral Overview. Quadrilateral Properties. Proof - Opposite Sides of Parallelogram Congruent. Proof - Diagonals of a Parallelogram Bisect Each Other. Proof - Opposite Angles of Parallelogram Congruent. Proof - Rhombus Diagonals are Perpendicular Bisectors. Proof - Rhombus Area Half Product of Diagonal Length. Area of a Parallelogram. Whether a Special Quadrilateral Can Exist. Rhombus Diagonals. Quadrilateral Overview. Quadrilateral Properties. Proof - Opposite Sides of Parallelogram Congruent. Proof - Diagonals of a Parallelogram Bisect Each Other. Proof - Opposite Angles of Parallelogram Congruent. Proof - Rhombus Diagonals are Perpendicular Bisectors. Proof - Rhombus Area Half Product of Diagonal Length. Area of a Parallelogram. Whether a Special Quadrilateral Can Exist. Rhombus Diagonals.

Language and Notation of the Circle. Circles: Radius, Diameter and Circumference. Length of an arc that subtends a central angle. Finding central angle measure given arc length. Parts of a Circle. Area of a Circle. Area of a sector given a central angle. Inscribed and Central Angles. Perpendicular Radius Bisects Chord. Right Triangles Inscribed in Circles (Proof). Area of Inscribed Equilateral Triangle (some basic trig used). Language and Notation of the Circle. Circles: Radius, Diameter and Circumference. Length of an arc that subtends a central angle. Finding central angle measure given arc length. Parts of a Circle. Area of a Circle. Area of a sector given a central angle. Inscribed and Central Angles. Perpendicular Radius Bisects Chord. Right Triangles Inscribed in Circles (Proof). Area of Inscribed Equilateral Triangle (some basic trig used).

This topic introduces the basic conceptual tools that underpin our journey through Euclidean geometry. These include the ideas of points, lines, line segments, rays, and planes. Euclid as the Father of Geometry. Language and Notation of Basic Geometry. Lines, Line Segments, and Rays. Recognizing rays lines and line segments. Specifying planes in three dimensions. Points, lines, and planes. Language and Notation of the Circle. The Golden Ratio. Identifying Rays. Measuring segments. Measuring segments. Congruent segments. Congruent segments. Segment addition. Segment addition. Algebraic midpoint of a segment exercise. Midpoint of a segment. Euclid as the Father of Geometry. Language and Notation of Basic Geometry. Lines, Line Segments, and Rays. Recognizing rays lines and line segments. Specifying planes in three dimensions. Points, lines, and planes. Language and Notation of the Circle. The Golden Ratio. Identifying Rays. Measuring segments. Measuring segments. Congruent segments. Congruent segments. Segment addition. Segment addition. Algebraic midpoint of a segment exercise. Midpoint of a segment.

We've always been communicating.... as we moved from signal fires, to alphabets & electricity the problems remained the same. What is Information Theory?. Prehistory: Proto-writing. Ptolemaic: Rosetta Stone. Ancient History: The Alphabet. Source Encoding. Visual Telegraphs (case study). Decision Tree Exploration. Electrostatic Telegraphs (case study). The Battery & Electromagnetism. Morse Code & The Information Age. Morse code Exploration. What's Next?. Symbol Rate. Symbol Rate Exploration. Introduction to Channel Capacity. Message Space Exploration. Measuring Information. Galton Board Exploration. Origin of Markov Chains. Markov Chain Exploration. A Mathematical Theory of Communication. Markov Text Exploration. What's Next?. What is Information Theory?. Prehistory: Proto-writing. Ptolemaic: Rosetta Stone. Ancient History: The Alphabet. Source Encoding. Visual Telegraphs (case study). Decision Tree Exploration. Electrostatic Telegraphs (case study). The Battery & Electromagnetism. Morse Code & The Information Age. Morse code Exploration. What's Next?. Symbol Rate. Symbol Rate Exploration. Introduction to Channel Capacity. Message Space Exploration. Measuring Information. Galton Board Exploration. Origin of Markov Chains. Markov Chain Exploration. A Mathematical Theory of Communication. Markov Text Exploration. What's Next?.

Let's get our feet wet by thinking in terms of vectors and spaces. Introduction to Vectors. Vector Examples. Scaling vectors. Adding vectors. Parametric Representations of Lines. Linear Combinations and Span. Introduction to Linear Independence. More on linear independence. Span and Linear Independence Example. Linear Subspaces. Basis of a Subspace. Vector Dot Product and Vector Length. Proving Vector Dot Product Properties. Proof of the Cauchy-Schwarz Inequality. Vector Triangle Inequality. Defining the angle between vectors. Defining a plane in R3 with a point and normal vector. Cross Product Introduction. Proof: Relationship between cross product and sin of angle. Dot and Cross Product Comparison/Intuition. Vector Triple Product Expansion (very optional). Normal vector from plane equation. Point distance to plane. Distance Between Planes. Matrices: Reduced Row Echelon Form 1. Matrices: Reduced Row Echelon Form 2. Matrices: Reduced Row Echelon Form 3. Matrix Vector Products. Introduction to the Null Space of a Matrix. Null Space 2: Calculating the null space of a matrix. Null Space 3: Relation to Linear Independence. Column Space of a Matrix. Null Space and Column Space Basis. 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. Showing relation between basis cols and pivot cols. Showing that the candidate basis does span C(A). Introduction to Vectors. Vector Examples. Scaling vectors. Adding vectors. Parametric Representations of Lines. Linear Combinations and Span. Introduction to Linear Independence. More on linear independence. Span and Linear Independence Example. Linear Subspaces. Basis of a Subspace. Vector Dot Product and Vector Length. Proving Vector Dot Product Properties. Proof of the Cauchy-Schwarz Inequality. Vector Triangle Inequality. Defining the angle between vectors. Defining a plane in R3 with a point and normal vector. Cross Product Introduction. Proof: Relationship between cross product and sin of angle. Dot and Cross Product Comparison/Intuition. Vector Triple Product Expansion (very optional). Normal vector from plane equation. Point distance to plane. Distance Between Planes. Matrices: Reduced Row Echelon Form 1. Matrices: Reduced Row Echelon Form 2. Matrices: Reduced Row Echelon Form 3. Matrix Vector Products. Introduction to the Null Space of a Matrix. Null Space 2: Calculating the null space of a matrix. Null Space 3: Relation to Linear Independence. Column Space of a Matrix. Null Space and Column Space Basis. 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. Showing relation between basis cols and pivot cols. Showing that the candidate basis does span C(A).

Understanding how we can map one set of vectors to another set. Matrices used to define linear transformations. A more formal understanding of functions. 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. More on Matrix Addition and Scalar Multiplication. Linear Transformation Examples: Scaling and Reflections. Linear Transformation Examples: Rotations in R2. Rotation in R3 around the X-axis. Unit Vectors. Introduction to Projections. Expressing a Projection on to a line as a Matrix Vector prod. Compositions of Linear Transformations 1. Compositions of Linear Transformations 2. 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. Example of 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. (correction) scalar multiplication of row. 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. Transpose of a Matrix Product. Transposes of sums and inverses. Transpose of a Vector. Rowspace and Left Nullspace. Visualizations of Left Nullspace and Rowspace. Rank(A) = Rank(transpose of A). Showing that A-transpose x A is invertible. A more formal understanding of functions. 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. More on Matrix Addition and Scalar Multiplication. Linear Transformation Examples: Scaling and Reflections. Linear Transformation Examples: Rotations in R2. Rotation in R3 around the X-axis. Unit Vectors. Introduction to Projections. Expressing a Projection on to a line as a Matrix Vector prod. Compositions of Linear Transformations 1. Compositions of Linear Transformations 2. 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. Example of 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. (correction) scalar multiplication of row. 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. Transpose of a Matrix Product. Transposes of sums and inverses. Transpose of a Vector. Rowspace and Left Nullspace. Visualizations of Left Nullspace and Rowspace. Rank(A) = Rank(transpose of A). Showing that A-transpose x A is invertible.

We explore creating and moving between various coordinate systems. Orthogonal Complements. 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. Rowspace Solution to Ax=b example. Projections onto Subspaces. Visualizing a projection onto a plane. A Projection onto a Subspace is a Linear Transforma. Subspace Projection Matrix Example. Another Example of a Projection Matrix. Projection is closest vector in subspace. Least Squares Approximation. Least Squares Examples. Another Least Squares Example. 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 Example. Alternate Basis Transformation Matrix Example Part 2. 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. Finding projection onto subspace with orthonormal basis example. Example using orthogonal change-of-basis matrix to find transformation matrix. Orthogonal matrices preserve angles and lengths. The Gram-Schmidt Process. Gram-Schmidt Process Example. Gram-Schmidt example with 3 basis vectors. Introduction to Eigenvalues and Eigenvectors. Proof of formula for determining Eigenvalues. Example solving for the eigenvalues of a 2x2 matrix. Finding Eigenvectors and Eigenspaces example. Eigenvalues of a 3x3 matrix. Eigenvectors and Eigenspaces for a 3x3 matrix. Showing that an eigenbasis makes for good coordinate systems. Orthogonal Complements. 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. Rowspace Solution to Ax=b example. Projections onto Subspaces. Visualizing a projection onto a plane. A Projection onto a Subspace is a Linear Transforma. Subspace Projection Matrix Example. Another Example of a Projection Matrix. Projection is closest vector in subspace. Least Squares Approximation. Least Squares Examples. Another Least Squares Example. 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 Example. Alternate Basis Transformation Matrix Example Part 2. 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. Finding projection onto subspace with orthonormal basis example. Example using orthogonal change-of-basis matrix to find transformation matrix. Orthogonal matrices preserve angles and lengths. The Gram-Schmidt Process. Gram-Schmidt Process Example. Gram-Schmidt example with 3 basis vectors. Introduction to Eigenvalues and Eigenvectors. Proof of formula for determining Eigenvalues. Example solving for the eigenvalues of a 2x2 matrix. Finding Eigenvectors and Eigenspaces example. Eigenvalues of a 3x3 matrix. Eigenvectors and Eigenspaces for a 3x3 matrix. Showing that an eigenbasis makes for good coordinate systems.

Using definite integrals with the shell and disc methods to find volumes of solids of revolution. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2. Disk method around x-axis. Generalizing disc method around x-axis. Disc method around y-axis. Disc method (washer method) for rotation around x-axis. Generalizing the washer method. Disc method rotation around horizontal line. Washer method rotating around non-axis. Part 2 of washer for non axis rotation. Disc method rotating around vertical line. Calculating integral disc method around vertical line. Washer or ring method for vertical line rotation. Evaluating integral for washer method around vertical line. Shell method for rotating around vertical line. Evaluating integral for shell method example. Shell method for rotating around horizontal line. Shell method with two functions of x. Calculating integral with shell method. Shell method with two functions of y. Part 2 of shell method with 2 functions of y. Disc method: function rotated about x-axis. Disc method (rotating f(x) about x axis). Volume of a sphere. Disc method with outer and inner function boundaries. Shell method to rotate around y-axis. Disk method: rotating x=f(y) around the y-axis. Shell method around a non-axis line. Shell method around a non-axis line 2.

Limit introduction, squeeze theorem, and epsilon-delta definition of limits. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2. Introduction to limits. Limit at a point of discontinuity. Determining which limit statements are true. Limit properties. Limit example 1. Limits 1. One-sided limits from graphs. One-sided limits from graphs. Introduction to Limits. Limit Examples (part 1). Limit Examples (part 2). Limit Examples (part 3). Limit Examples w/ brain malfunction on first prob (part 4). More Limits. Limits 1. Limits and infinity. Limits at positive and negative infinity. More limits at infinity. Limits with two horizontal asymptotes. Limits 2. Squeeze Theorem. Proof: lim (sin x)/x. Limit intuition review. Building the idea of epsilon-delta definition. Epsilon-delta definition of limits. Proving a limit using epsilon-delta definition. Limits to define continuity. Continuity. Epsilon Delta Limit Definition 1. Epsilon Delta Limit Definition 2.

Divergence theorem intuition. Divergence theorem examples and proofs. Types of regions in 3D. 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5). 3-D Divergence Theorem Intuition. Divergence Theorem Example 1. Why we got zero flux in Divergence Theorem Example 1. Type I Regions in Three Dimensions. Type II Regions in Three Dimensions. Type III Regions in Three Dimensions. Divergence Theorem Proof (part 1). Divergence Theorem Proof (part 2). Divergence Theorem Proof (part 3). Divergence Theorem Proof (part 4). Divergence Theorem Proof (part 5).

This topic continues our journey through the world of Euclid by helping us understand angles and how they can relate to each other. Angle basics. Measuring angles in degrees. Using a protractor. Measuring angles. Measuring angles. Acute right and obtuse angles. Angle types. Vertical, adjacent and linearly paired angles. Exploring angle pairs. Introduction to vertical angles. Vertical angles. Using algebra to find the measures of vertical angles. Vertical angles 2. Proof-Vertical Angles are Equal. Angles Formed by Parallel Lines and Transversals. Identifying Parallel and Perpendicular Lines. Figuring out angles between transversal and parallel lines. Congruent angles. Parallel lines 1. Using algebra to find measures of angles formed from transversal. Parallel lines 2. CA Geometry: Deducing Angle Measures. Proof - Sum of Measures of Angles in a Triangle are 180. Triangle Angle Example 1. Triangle Angle Example 2. Triangle Angle Example 3. Challenging Triangle Angle Problem. Proof - Corresponding Angle Equivalence Implies Parallel Lines. Finding more angles. Angles 1. Angles 2. Sum of Interior Angles of a Polygon. Angles of a polygon. Sum of the exterior angles of convex polygon. Introduction to angles (old). Angles (part 2). Angles (part 3). Angles formed between transversals and parallel lines. Angles of parallel lines 2. The Angle Game. Angle Game (part 2). Acute right and obtuse angles. Complementary and supplementary angles. Complementary and supplementary angles. Example using algebra to find measure of complementary angles. Example using algebra to find measure of supplementary angles. Angle addition postulate. Angle basics. Measuring angles in degrees. Using a protractor. Measuring angles. Measuring angles. Acute right and obtuse angles. Angle types. Vertical, adjacent and linearly paired angles. Exploring angle pairs. Introduction to vertical angles. Vertical angles. Using algebra to find the measures of vertical angles. Vertical angles 2. Proof-Vertical Angles are Equal. Angles Formed by Parallel Lines and Transversals. Identifying Parallel and Perpendicular Lines. Figuring out angles between transversal and parallel lines. Congruent angles. Parallel lines 1. Using algebra to find measures of angles formed from transversal. Parallel lines 2. CA Geometry: Deducing Angle Measures. Proof - Sum of Measures of Angles in a Triangle are 180. Triangle Angle Example 1. Triangle Angle Example 2. Triangle Angle Example 3. Challenging Triangle Angle Problem. Proof - Corresponding Angle Equivalence Implies Parallel Lines. Finding more angles. Angles 1. Angles 2. Sum of Interior Angles of a Polygon. Angles of a polygon. Sum of the exterior angles of convex polygon. Introduction to angles (old). Angles (part 2). Angles (part 3). Angles formed between transversals and parallel lines. Angles of parallel lines 2. The Angle Game. Angle Game (part 2). Acute right and obtuse angles. Complementary and supplementary angles. Complementary and supplementary angles. Example using algebra to find measure of complementary angles. Example using algebra to find measure of supplementary angles. Angle addition postulate.

Conditional statements and deductive reasoning. Conditional statements exercise examples. Logical argument and deductive reasoning exercise example. Conditional statements and deductive reasoning. Conditional statements exercise examples. Logical argument and deductive reasoning exercise example.

2003 AIME II Problem 1. 2003 AIME II Problem 3. Sum of factors of 27000. Sum of factors 2. 2003 AIME II Problem 4 (part 1). 2003 AIME II Problem 4 (part 2). 2003 AIME II Problem 5. 2003 AIME II Problem 5 Minor Correction. Area Circumradius Formula Proof. 2003 AIME II Problem 6. 2003 AIME II Problem 7. 2003 AIME II Problem 8. Sum of Polynomial Roots (Proof). Sum of Squares of Polynomial Roots. 2003 AIME II Problem 9. 2003 AIME II Problem 10. 2003 AIME II Problem 11. 2003 AIME II Problem 12. 2003 AIME II Problem 13. 2003 AIME II Problem 14. 2003 AIME II Problem 15 (part 1). 2003 AIME II Problem 15 (part 2). 2003 AIME II Problem 15 (part 3). 2003 AIME II Problem 1. 2003 AIME II Problem 3. Sum of factors of 27000. Sum of factors 2. 2003 AIME II Problem 4 (part 1). 2003 AIME II Problem 4 (part 2). 2003 AIME II Problem 5. 2003 AIME II Problem 5 Minor Correction. Area Circumradius Formula Proof. 2003 AIME II Problem 6. 2003 AIME II Problem 7. 2003 AIME II Problem 8. Sum of Polynomial Roots (Proof). Sum of Squares of Polynomial Roots. 2003 AIME II Problem 9. 2003 AIME II Problem 10. 2003 AIME II Problem 11. 2003 AIME II Problem 12. 2003 AIME II Problem 13. 2003 AIME II Problem 14. 2003 AIME II Problem 15 (part 1). 2003 AIME II Problem 15 (part 2). 2003 AIME II Problem 15 (part 3).