Linear Combinations: Definition & Equation

Linear Combinations: Definition & Equation
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  • 0:04 Practical Example
  • 1:35 Official Definitions
  • 4:01 Equation for Linear…
  • 4:40 Math Example
  • 5:27 Lesson Summary
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Lesson Transcript
Instructor: Jenna McDanold

Jenna has two master's degrees in mathematics and has been teaching as an adjunct professor in Chicago for four years.

In this lesson, you will learn about linear combinations by using a practical example. We will define linear combinations and their related terms using mathematical terminology and show another example.

Practical Example

Francine and Fred are attending a wedding reception, and they're offering a buffet of delicious food. The buffet includes a fruit tray, a vegetable tray, a cheese tray, a meatball tray, and a dessert truffle tray. Each guest fills a plate; some will take only one item; others will take several items. Francine takes 1 piece of fruit, 5 vegetables, 2 pieces of cheese, 1 meatball, and 2 truffles. Fred takes 3 pieces of fruit, 4 pieces of cheese, and 2 meatballs.

When they make a plate, they're taking a multiple of each tray. Since the items from each tray are not cut into smaller pieces, the multiples are represented by the set of whole numbers: 0, 1, 2, 3, and so on. The entire plate is the sum of these multiples. Using mathematical notation, Francine's plate would be:

1 * fruit + 5 * vegetables + 2 * cheese + 1 * meatballs + 2 * truffles = (1 fruit, 5 veg, 2 cheeses, 1 meatball, 2 truffles)

Francine took at least one item from each tray, but Fred chose items from only three of the trays. We include 0 as a multiple, and therefore Fred's plate would be:

3 * fruits + 0 * vegetables + 4 * cheeses + 2 * meatballs + 0 * truffles = (3 fruits, 0 veg, 4 cheeses, 2 meatballs, 0 truffles)

We can call each plate a linear combination of the trays in the buffet over the set of whole numbers.

Official Definitions

Before moving on, let's go over some official definitions. A linear combination is the sum of scalar multiples of the vectors in a generating set. To understand this definition, we must explain some of the terms used within it.

A vector is an element of a set that contains at least two coordinates or more. Vectors can also contain an infinite amount of coordinates.

(1, 2) is an element from the vector space R2 that contains elements with two coordinates.

(1, 2, 3) is an element from the vector space R3 that contains elements with three coordinates.

The plates of food from the wedding example are vectors with five coordinates.

A generating set is a set that contains a collection of vectors that may create desired linear combinations. In the wedding example, the set {fruit, vegetables, cheese, meatballs, truffles} would be our generating set, or the set of all options offered at the buffet. We need to express this set as a set of vectors, each with five coordinates:

  • fruit = (1 fruit, 0 veg, 0 cheese, 0 meatballs, 0 truffles)
  • veg = (0 fruit, 1 veg, 0 cheese, 0 meatballs, 0 truffles)
  • cheese = (0 fruit, 0 veg, 1 cheese, 0 meatballs, 0 truffles)
  • meatballs = (0 fruit, 0 veg, 0 cheese, 1 meatball, 0 truffles)
  • truffles = (0 fruit, 0 veg, 0 cheese, 0 meatballs, 1 truffle)

The set that includes the five stated vectors is our generating set for the wedding reception.

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