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Vector Spaces: Definition & Example

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  • 0:00 What Are Vector Spaces?
  • 2:22 Axioms for Vector Spaces
  • 4:31 Example of Vector Spaces
  • 13:09 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, 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.

What Are Vector Spaces?

To define a vector space, first we need a few basic definitions. A set is a collection of distinct objects called elements. The elements are usually real or complex numbers when we use them in mathematics, but the elements of a set can also be a list of things. We notate a set by encasing the elements within curly braces. Note that to be distinct, an element cannot be repeated within the same set.

  • {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is the set of single-digit numbers that we use in mathematics.
  • {a, b, c, d, . . ., y, z} is the set of letters in the alphabet.

Let's now take a closer look at elements in vector spaces. First, it's important to note that a space in mathematics is a set in which the list of elements are defined by a collection of guidelines or axioms for how each element relates to another within the set.

A vector space is a space in which the elements are sets of numbers themselves. Each element in a vector space is a list of objects that has a specific length, which we call vectors. We usually refer to the elements of a vector space as n-tuples, with n as the specific length of each of the elements in the set.

Each element of a vector space of length n can be represented as a matrix, which you may recall is a collection of numbers within parentheses. Matrix representations require multiple other lessons in matrix multiplication and addition, so we will use the parentheses notation for this lesson.

Here's an example: In the 4-dimensional vector space of the real numbers, notated as R4, one element is (0, 1, 2, 3). This vector has four parts and is a single element within the vector space R4.

Now let's take a closer look at fields. We refer to any vector space as a vector space defined over a given field F. A field is a space of individual numbers, usually real or complex numbers. The specific axioms to define a field are similar to those of a vector space, so for the purposes of this lesson, we'll define a field as a vector space whose elements are single numbers that adhere to the same set of axioms as those we are about to explain.

Axioms for Vector Spaces

There are ten axioms that define a vector space. We let x, y, and z be elements of the vector space V. We let a and b be elements of the field F.

  1. Closed under addition: For each element x and y in V, x + y is also in V.
  2. Closed under scalar multiplication: For each element x in V and scalar a in F, ax is in V.
  3. Commutativity of addition: For each element x and y in V, x + y = y + x.
  4. Associativity of addition: For each element x, y, and z in V, (x + y) + z = x + (y + z).
  5. Existence of the additive identity: There exists an element in V denoted as 0 such that x + 0 = x, for all x in V.
  6. Existence of the additive inverse: For each element x in V, there exists another element in V that we will call -x such that x + (-x) = 0.
  7. Existence of the multiplicative identity: There exists an element in F notated as 1 such that for all x in V, 1x = x.
  8. Associativity of scalar multiplication: For each element x in V, and for each pair of elements a and b in F, (ab)x = a(bx).
  9. Distribution of elements to scalars: For each element a in F and each pair of elements x and y in V, a(x + y) = ax + ay.
  10. Distribution of scalars to elements: For each element x in V, and each pair of elements a and b in F, (a + b)x = ax + bx.

Example of Vector Spaces

These are the spaces of n-tuples in which each part of each element is a real number, and the set of scalars is also the set of real numbers. Let's take a closer look at some key definitions for our example.

  • Addition is defined as adding the corresponding parts of each element: (a, b, . . . ) + (c, d, . . . ) = (a + c, b + d, . . .).
  • Scalar multiplication is defined as multiplying every part of the element by the scalar: a(b, c, . . . ) = (ab, ac, . . . ).
  • The additive identity for these vector spaces is the element (0, 0, 0, . . . , 0), in which there are n 0s in this element.
  • The multiplicative identity for these vector spaces is the scalar 1 from the field of real numbers R.

The following is a basic example, but not a proof that the space R3 is a vector space.

Axiom 1: Closure of Addition

Let x = (0, 1, 2), and let y = (3, 4, 5) from R3:

  • x + y = (0, 1, 2) + (3, 4, 5) = (0 + 3, 1 + 4, 2 + 5) = (3, 5, 7)

Another element in R3 is (3, 5, 7). We can generalize the above as follows:

  • x + y = (a, b, c) + (d, e, f) = (a + d, b + e, c + f)

(a + d, b + e, c + f) is another element of R3, and therefore this space is closed under addition.

Axiom 2: Closure of Scalar Multiplication

Let x be (a, b, c) from R3, and let g be an element from the field F.

  • g(a, b, c) = (ga, gb, gc)

Since (ga, gb, gc) is another element of R3, this space is also closed under scalar multiplication.

Axiom 3: Commutativity of Addition

Let x = (0, 1, 2), and let y = (3, 4, 5) from R3:

  • x + y = (0, 1, 2) + (3, 4, 5) = (0 + 3, 1 + 4, 2 + 5) = (3, 5, 7)
  • y + x = (3, 4, 5) + (0, 1, 2) = (3 + 0, 4 + 1, 5 + 2) = (3, 5, 7)

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