This 12-minute video lesson finds that the orthogonal complement of the orthogonal complement of V is V. [Linear Algebra playlist: Lesson 103 of 143]

This 11-minute video lesson shows that orthogonal matrices preserve angles and lengths. [Linear Algebra playlist: Lesson 129 of 143]

This 11-minute video lesson proves the "associative," "distributive," and "commutative" properties for vector dot products.

This 23-minute video lesson looks at Rowspace and Left Nullspace. [Linear Algebra playlist: Lesson 97 of 143]

This 9-minute video lesson provides definitions of the vector dot product and vector length.

This 20-minute video lesson looks at the relationship between left nullspace, rowspace, column space and nullspace. [Linear Algebra playlist: Lesson 98 of 143]

What were the causes and effects of westward expansion between 1844 and 1877? Kim Kutz Elliott discusses how economic opportunities, government support, and the idea of 'manifest destiny' brought migrants to the western United States.

This lesson discusses how to identify sets of numbers as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

This site teaches Quantities to High Schoolers through a series of 116 questions and interactive activities aligned to 4 Common Core mathematics skills.

This site teaches The Complex Number System to High Schoolers through a series of 2195 questions and interactive activities aligned to 16 Common Core mathematics skills.

This site teaches The Real Number System to High Schoolers through a series of 1090 questions and interactive activities aligned to 8 Common Core mathematics skills.

This site teaches Vector and Matrix Quantities to High Schoolers through a series of 2195 questions and interactive activities aligned to 16 Common Core mathematics skills.

To plot an inequality, such as x>3, on a number line, first draw a circle over the number (e.g., 3). Then if the sign includes equal to (≥ or ≤), fill in the circle. If the sign does not include equal to (> or <), leave the circle unfilled in. Finally, draw a line going from the circle in the direction of the numbers that make the inequality true.

This 15-minute video lesson gives an example involving the preimage of a set under a transformation. It also gives a definition of kernel of a transformation.

Khan academy video with clear visual lecture on the main features of primaries and caucuses.