You will learn about linear independence, linear dependence, and linear combinations of vectors.
After completing this tutorial, you will be able to complete the following:
Recall that vectors are quantities consisting of direction and magnitude. They are represented in the plane by directed line segments, and written algebraically using Cartesian coordinates. First, consider a car that travels 50 miles due north, turns, and then travels east for 50 more miles. This situation is depicted using vectors below:
At the end of its journey, the car stops at point (50, 50), which is approximately 70.71 miles from the car's starting point and in the northeast direction. This is a vector quantity, since it consists of both length (approximately 70.71 miles) and direction (northeast). While it is true that the car is approximately 70.71 miles away from its starting point and in the northeast direction, this is not necessary information. A person would know the location of the car just by knowing that the car traveled 50 miles north and then 50 miles east.
In this example, the "50 miles north" vector and the "50 miles east" vector are linearly independent-neither vector can be described in terms of the other. The "70.71 miles northeast" vector is a linear combination of the other two vectors-it can be written as the sum of scalar multiples of the two vectors (in this case the scalars are both 1). Thus, the set of three vectors together is said to be linearly dependent because one of the vectors is superfluous.
We can formalize these ideas in the following definitions:
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know scalar multiplication of a vector by a real number and the addition of vectors.|
|Type of Tutorial||Concept Development|
|Key Vocabulary||linear combination, linear dependence of vectors, linear independence of vectors|