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ZingPath: Inner Product and Linear Dependence

Linear Dependence of Vectors

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Inner Product and Linear Dependence

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Linear Dependence of Vectors


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You will learn about linear independence, linear dependence, and linear combinations of vectors.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Know the definition of linear dependence and independence of two vectors.
  • Determine whether any two given vectors are linearly dependent or independent.
  • Understand that three distinct vectors in the plane must be linearly dependent.
  • Know that any vector in the plane can be written as a linear combination of a given pair of linearly independent vectors.

Everything You'll Have Covered

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:

Tutorial Details

Approximate Time 20 Minutes
Pre-requisite Concepts Students should know scalar multiplication of a vector by a real number and the addition of vectors.
Course Algebra-2
Type of Tutorial Concept Development
Key Vocabulary linear combination, linear dependence of vectors, linear independence of vectors