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Mathematics LibreTexts

5: Vector Spaces

  2. Last updated
    Mar 5, 2021
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( \newcommand{\kernel}{\mathrm{null}\,}\)

The two key properties of vectors are that they can be added together and multiplied by scalars, so we make the following definition.

Definition

A vector space (V,+,.,R) is a set V with two operations + and ⋅ satisfying the following properties for all u,v∈V and c,d∈R:

  1. (Additive Closure) u+v∈V. Adding two vectors gives a vector.
  2. (Additive Commutativity) u+v=v+u. Order of addition doesn’t matter.
  3. (Additive Associativity) (u+v)+w=u+(v+w). Order of adding many vectors doesn’t matter.
  4. (Zero) There is a special vector 0V∈V such that u+0V=u for all u in V.
  5. (Additive Inverse) For every u∈V there exists w∈V such that u+w=0V.
  6. (Multiplicative Closure) c⋅v∈V. Scalar times a vector is a vector.
  7. (Distributivity) (c+d)⋅v=c⋅v+d⋅v. Scalar multiplication distributes over addition of scalars.
  8. (Distributivity) c⋅(u+v)=c⋅u+c⋅v. Scalar multiplication distributes over addition of vectors.
  9. (Associativity) (c⁢d)⋅v=c⋅(d⋅v).
  10. (Unity) 1⋅v=v for all v∈V.

Remark

Rather than writing (V,+,.,R), we will often say "let V be a vector space over R''. If it is obvious that the numbers used are real numbers, then "let V be a vector space'' suffices. Also, don't confuse the scalar product with the dot product. The scalar product is a function that takes as inputs a number and a vector and returns a vector as its output. This can be written:

(5.1)⋅:R×V→V.

Similarly

(5.2)+:V×V→V.

On the other hand, the dot product takes two vectors and returns a number. Succinctly: ⋅:V×V→ℜ. Once the properties of a vector space have been verified, we'll just write scalar multiplication with juxtaposition c⁢v=c⋅v, though, to avoid confusing the notation.

  • 5.1: Examples of Vector Spaces
    One can find many interesting vector spaces, such as the following:
  • 5.2: Other Fields
    Above, we defined vector spaces over the real numbers. One can actually define vector spaces over any field. This is referred to as choosing a different base field. A field is a collection of "numbers'' satisfying certain properties.
  • 5.3: Review Problems

Contributor

This page titled 5: Vector Spaces is shared under a not declared license and was authored, remixed, and/or curated by David Cherney, Tom Denton, & Andrew Waldron.

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