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Math

Algebra II

Conceptual videos and worked examples from basic algebra through algebra 2. Includes videos from the former algebra worked examples playlists.
Community Questions
A thumbnail for: Systems of equations and inequalities
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Systems of equations and inequalities

What happens when we have many variables but also many constraints.
A thumbnail for: Functions and their graphs
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Functions and their graphs

Revisiting what a function is and how we can define and visualize one.
A thumbnail for: Polynomial and rational functions
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Polynomial and rational functions

Exploring quadratics and higher degree polynomials. Also in-depth look at rational functions.
A thumbnail for: Rational expressions
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Rational expressions

You have probably been wondering whether our powers of algebraic problem solving break down if we divide by the variable or we have entire expressions in denominator of a fraction. Well, they don't! In this topic, you'll learn how to interpret and manipulate rational expressions (when you have one algebraic expression divided by another)!
A thumbnail for: Exponential and logarithmic functions
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Exponential and logarithmic functions

An look at exponential and logarithmic functions including many of their properties and graphs.
A thumbnail for: Logarithms
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Logarithms

Log-a-what? No, this tutorial is about neither chopped wood nor music (actually logarithms do have applications in music), but it is fascinating nonetheless. You know how to take an exponent. Now you can think about what exponent you have to raise a number to to get another number. Yes, I agree--unstoppable fun for the whole family. No, seriously, logarithms are used everywhere (including to measure earthquakes and sound).
A thumbnail for: Imaginary and complex numbers
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Imaginary and complex numbers

Understanding and solving equations with imaginary numbers.
A thumbnail for: Conic sections
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Conic sections

A detailed look at shapes that are prevalent in science: conic sections
A thumbnail for: Matrices
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Matrices

Understanding and solving matrices.
Matrices
Understanding and solving matrices.

Basic matrix operations

Keanu Reeves' virtual world in the The Matrix (I guess we can call all three movies "The Matrices") have more in common with this tutorial than you might suspect. Matrices are ways of organizing numbers. They are used extensively in computer graphics, simulations and information processing in general. The super-intelligent artificial intelligences that created The Matrix for Keanu must have used many matrices in the process. This tutorial introduces you to what a matrix is and how we define some basic operations on them.

Matrix multiplication

You know what a matrix is, how to add them and multiply them by a scalar. Now we'll define multiplying one matrix by another matrix. The process may seem bizarre at first (and maybe even a little longer than that), but there is a certain naturalness to the process. When you study more advanced linear algebra and computer science, it has tons of applications (computer graphics, simulations, etc.)

Properties of matrix multiplication

As we'll see, some of the properties of scalar multiplication (like the distributive and associative properties) have analogs in matrix multiplication while some don't (the commutative property).

Zero and identity matrices

In arithmetic, we learned than a number times 1 is still that number and that anything times 0 is 0. In this tutorial, we attempt to extend these ideas to the world of matrices!

Geometric transformations with matrices

We'll now see one of the most powerful applications of matrix multiplication--geometric transformations. This is core of what videos games and computer-based animation uses to "transform" figures based on movement or perspective. You probably never thought matrices could be so much fun!

Reduced row echelon form

You've probably already appreciated that there are many ways to solve a system of equations. Well, we'll introduce you to another one in this tutorial. Reduced row echelon form has us performing operations on matrices to get them in a form that helps us solve the system.