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Linear Combinations & Span: Definition & Equation

Instructor: Jenna McDanold

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

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.

Practical Example

Imagine that you have three fields of fruit: apples, oranges, and pears. At any given time, you have a total of 500 sellable pieces of each type of fruit. You are selling these fruits in mixed bushels.

Now consider how many different mixes you could create. Starting with a bushel with one piece of fruit to a bushel with hundreds of pieces of fruit, you record all of the possible bushels. Some of the bushels in the list would be:

  • 1 apple
  • 1 orange
  • 1 pear
  • 1 apple, 1 pear
  • 1 orange, 1 pear
  • 1 apple, 1 orange, 1 pear
  • 2 apples
  • 2 apples, 5 pears
  • ...
  • 500 apples, 2 oranges, 499 pears
  • ...
  • 500 apples, 500 oranges, 500 pears

Notice that you are multiplying each piece of available fruit by a specific number between 0 and 500, and then adding those multiples together to create each bushel description.

  • 1 apple = (1 * 1 apple) + (0 * 1 orange) + (0 * 1 pear)
  • 1 apple, 1 orange = (1 * 1 apple) + (1 * 1 orange) + (0 * 1 pear)
  • ...
  • 500 apples, 2 oranges, 499 pears = (500 * 1 apple) + (2 * 1 orange) + (499 * 1 pear)

In mathematical terms, we would call the set {1 apple, 1 orange, 1 pear} the 'basis' set for this example, since this a simple set that includes one of each type of fruit.

Applicable Definitions

Since you have a maximum of 500 pieces of each fruit, we will call the set {0, 1, 2, ... 500} the set of scalars. Scalars are the numbers that you may multiply each element of your basis set by to find a new combination of your elements.

Each bushel description is called a linear combination of the pieces of fruit over the set of numbers from 0 to 500. The entire list of bushel descriptions is called the span of the set of fruit over the set of numbers.

Official Definitions

For both of these definitions, we must have the following:

  • Let V be a vector space.
    • In our example, this would be the set of all possible fruit that you could grow.
  • Let B be a nonempty subset of V that includes the most basic elements from the vector space.
    • This is the set {1 apple, 1 orange, 1 pear}, which is a subset of the vector space of possible fruits.
  • Let {b1, b2, b3, ..., bm} be a list of the vectors in B.
    • In the format of linear combinations, this is the set: {(1 apple, 0 oranges, 0 pears), (0 apples, 1 orange, 0 pears), (0 apples, 0 oranges, 1 pear)}
  • Let F be a field, and S be a subset of that field.
    • This is the set {0, 1, 2, ..., 500} which is a subset of the field of real numbers.
  • Now let {a1, a2, a3, ..., an} be a subset of scalars from S
    • This list could be {500, 2, 499} for the bushel description for 500 apples, 2 oranges, and 499 pears. This is a subset of our field {0, 1, 2, ..., 500}.

Now our core definitions:

  • The vector v from our vector space V is a linear combination if v = a1b1 + a2b2 + ... + anbn
    • Each bushel description above is a linear combination of the subset of available fruit and a subset of the set {0, 1, 2, ..., 500}.
  • The span of the subset B of V denoted 'span(B)' is the set of all of the linear combinations of the vectors in the set B over the scalars in the field F.
    • The entire list of bushel descriptions is the span of the available fruit over the set {0, 1, 2, ...500}.

Math Example

Similarly, consider the vector space R3. This is the set of all triplet vectors of real numbers, (a, b, c).

Now we take the field of real numbers as our scalar set.

There are three elements to the vector space R3 that we can use for our basis set, B:

  • (1, 0, 0),
  • (0, 1, 0), and
  • (0, 0, 1)

This is the simplest set that we can achieve with each coordinate accounted for, just like the set outlined above with fruit as units for each of these numbers.

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