About This Chapter
Below is a sample breakdown of the Vectors in Linear Algebra chapter into a 5-day school week. Based on the pace of your course, you may need to adapt the lesson plan to fit your needs.
|Day||Topics||Key Terms and Concepts Covered|
|Monday||Vectors||Scalars, vectors, math operations on vectors, and dot products|
|Tuesday||Vector Spaces||Definition of and operations using vector spaces/subspaces and vector space basis|
|Wednesday||Orthonormal bases||Working with orthonormal bases and vectors, including the Gram-Schmidt process|
|Thursday||Linear combinations||How to use linear combinations and linear span to solve for vectors|
|Friday||Linear independence||Define linear dependence/independence and provide applications of their use|
1. Scalars and Vectors: Definition and Difference
In this lesson, we will examine scalars and vectors, learn why it is important to know the difference between the two and why remembering to add a direction to many of your exam answers could be the reason you get it right or wrong.
2. Performing Operations on Vectors in the Plane
After watching this video lesson, you should be able to add, subtract, and multiply your vectors. Learn how easy it is to perform these operations and what you need to keep in mind when performing these operations.
3. The Dot Product of Vectors: Definition & Application
After watching this video lesson, you will be able to find the dot product of vectors both algebraically and geometrically. Learn the difference between the two and what you need in order to calculate them.
4. Vector Spaces: Definition & Example
In this lesson, we'll discuss the definition and provide some common examples of vector spaces. We'll go over set theory, the axioms for vector spaces, and examples of axioms using vector spaces of the real numbers over a field of real numbers.
5. Finding the Basis of a Vector Space
In this lesson we'll start by reviewing matrix reduced row echelon form, which is integral to finding a basis of a vector space. Then we'll work through a problem together to see exactly how finding a basis is accomplished.
6. Orthonormal Bases: Definition & Example
In this lesson we show how independent vectors in a space can become a basis for the space and how this basis can be turned into an orthonormal basis. Having an orthonormal basis is useful in many applications involving vectors.
7. The Gram-Schmidt Process for Orthonormalizing Vectors
Linearly combining things is something we do quite naturally. When the things are vectors, there is a fantastic way to organize the vectors before combining them. In this lesson, we'll show how to orthonormalize vectors using the Gram-Schmidt process.
8. Linear Combinations & Span: Definition & Equation
This lesson will cover the definitions of linear combinations and spans in terms of vector spaces, using a real world example and then a mathematical example. You will learn the official definitions and how to apply them in mathematics.
9. Linear Dependence & Independence: Definition & Examples
Linear dependence and independence are based on whether or not there is more than one solution to a system of equations. In this lesson, we'll look at how you can determine whether or not a system is independent and work through some examples.
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