The notion of vector space

Vector spaces: Vector spaces and linear subspaces

The notion of vector space

In the chapter Vector calculus in plane and space we got acquainted with vectors in the plane and in space. We also learned how to calculate with those: we can add them and perform scalar multiplication. When doing so, we use a number of obvious calculation rules, the most important of which are summarized below.

Properties of vector calculus

For all vectors #\vec{p}#, #\vec{q}#, #\vec{r}# and for all numbers (scalars) #\lambda#, #\mu#, the following eight rules hold:

Commutativity: #\vec{p} +\vec{q} =\vec{q}+\vec{p}#
Associativity of addition: #(\vec{p}+\vec{q})+\vec{r} =\vec{p}+(\vec{q}+\vec{r})#
Zero vector: There is a zero vector #\vec{0}# with the property #\vec{p} +\vec{0}=\vec{p}# for each #\vec{p}#
Negative: Every vector #\vec{p}# has a negative #-\vec{p}# such that #\vec{p} + -\vec{p}= \vec{0}# (we often simply write #\vec{p}-\vec{p}=\vec{0}#)
Scalar one: #1\cdot \vec{p}=\vec{p}#
Associativity of the scalar multiplication: # (\lambda \cdot\mu )\cdot\vec{p} =\lambda \cdot(\mu\cdot \vec{p})#
Distributivity of the scalar multiplication over the scalar addition: # (\lambda +\mu )\cdot\vec{p} =\lambda\cdot \vec{p} +\mu \cdot\vec{p}#
Distributivity of the scalar multiplication over vector addition: #\lambda\cdot (\vec{p} +\vec{q})=\lambda\cdot \vec{p} +\lambda\cdot \vec{q}#

The first four rules are found in Calculation rules for the addition of vectors. The fifth rule is found in the definition of scalars and the sixth rule is found in Calculation rules for scalar multiplication. The last two lines are formulated in Calculation rules for scalar multiplication and addition of vectors.

Because of associativity of addition we can formulate the expression in each part of the equation in 2. without brackets: #\vec{p}+\vec{q}+\vec{r}#.

Because of associativity of the scalar multiplication we can formulate the expression in each part of the equation in 6. without brackets: #\lambda \cdot\mu \cdot\vec{p}#.

The set #\mathbb{R}# of all real numbers, with the usual addition and multiplication satisfies all rules.

In this, the multiplication #2\cdot 3# is seen as the scalar multiplication of the scalar #2# with the vector #3#.

Next, we approach the concept of vector at a more abstract level.

Vector space
If we have a collection of objects that we can add with each other and for which a multiplication with numbers is defined such that the above eight calculation rules hold (the axioms of a vector space), then we the objects are called vectors and the set is called a vector space or linear space.

Vectors are often denoted with an arrow over a symbol, as in #\vec{v}# or #\vec{AB}#.

The numbers with which we multiply can be either real numbers or complex numbers. In the first case we speak of a real vector space and in the second of a complex vector space. These numbers are often called scalars (singular: scalar).

Other notations for #\vec{v}# that occur frequently in the literature are #\bar{v}#, #{\bf v}#, and #\underline{v}#.

Other sets of numbers than the real or complex numbers may also be used, but are beyond the scope of this course.

Although arrows in a plane #\mathbb{E}^2# or in the space #\mathbb{E}^3# each represent only one example of a vector space, it is often useful to support the intuition with figures of #\mathbb{E}^2# or #\mathbb{E}^3#.

The real numbers form a vector space with the usual addition and ordinary multiplication as scalar multiplication. In this exceptional case, the scalars coincide with the vectors.

From the eight calculation rules we can deduce more (very obvious) calculation rules.

Some simple consequences of the definition of vector space

In every vector space, the following rules are satisfied.

There is exactly one zero vector.
The negative of each vector is unique.
For each vector #\vec{u}# we have #0\cdot \vec{u} =\vec{0}#.
For each number #\lambda# we have #\lambda \cdot \vec{0}=\vec{0}#.
If a vector #\vec{u}# is distinct from the zero vector, and #\lambda# is a scalar such that #\lambda\cdot\vec{u}=\vec{0}#, then we have #\lambda=0#.

1. There is exactly one zero vector: suppose #\vec{0}# and #\vec{n}# are zero vectors. Then we have \[\begin{array}{rclll}\vec{n}&=&\vec{n} +\vec{0}&&\phantom{xx}\color{blue}{\vec{0}\text{ is a zero vector}}\\ &=&\vec{0} +\vec{n}&&\phantom{xx}\color{blue}{\text{commutativity}}\\ &=&\vec{0}&&\phantom{xx}\color{blue}{\vec{n}\text{ is a zero vector}}\end{array}\] So #\vec{n}=\vec{0}#, from which we conclude that the zero vector is unique.

2. The negative of each vector is unique: suppose that #-\vec{v}# and #\vec{u}# are both negative vectors of #\vec{v}#. Then we have \[\begin{array}{rclll}-\vec{v}&=&\left(-\vec{v}\right)+\vec{0} &&\phantom{xx}\color{blue}{\vec{0}\text{ is a zero vector}}\\ &=&\left(-\vec{v}\right) +\left(\vec{v}+\vec{u}\right)&&\phantom{xx}\color{blue}{\vec{u}\text{ is an opposite of }\vec{v}}\\ &=&\left(\left(-\vec{v}\right) +\vec{v}\right)+\vec{u}&&\phantom{xx}\color{blue}{\text{associativity}}\\ &=&\left(\vec{v}+\left(-\vec{v}\right) \right)+\vec{u}&&\phantom{xx}\color{blue}{\text{commutativity}}\\ &=&\vec{0}+\vec{u}&&\phantom{xx}\color{blue}{-\vec{v}\text{ is a negative of }\vec{v}}\\&=&\vec{u}+\vec{0}&&\phantom{xx}\color{blue}{\text{commutativity}}\\&=&\vec{u}&&\phantom{xx}\color{blue}{\vec{0}\text{ is a zero vector}}\end{array}\] So #-\vec{v}=\vec{u}#, from which we conclude that #\vec{v}# has exactly one negative.

3. The rule #0\cdot \vec{u}=\vec{0}# follows from \[\begin{array}{rclll}0\cdot \vec{u}&=&(1-1)\cdot\vec{u}&&\phantom{xx}\color{blue}{0=1-1}\\&=&\vec{u}-\vec{u}&&\phantom{xx}\color{blue}{\text{distributivity}}\\&=&\vec{0}&&\phantom{xx}\color{blue}{\text{negative vector}}\\\end{array}\]

4. The rule #\lambda\cdot\vec{0}=\vec{0}# follows from the previous rule if #\lambda=0#. Suppose therefore that #\lambda\ne0#. Then, for each vector #\vec{v}#: \[\begin{array}{rclll} \vec{v}+\lambda\cdot\vec{0}&=&1\cdot\vec{v}+\lambda\cdot\vec{0}&&\phantom{xx}\color{blue}{\text{scalar one}}\\&=&\lambda\cdot\lambda^{-1}\cdot\vec{v}+\lambda\cdot\vec{0}&&\phantom{xx}\color{blue}{\text{associativity}}\\&=&\lambda\cdot\left(\lambda^{-1}\cdot\vec{v}+\vec{0}\right)&&\phantom{xx}\color{blue}{\text{distributivity}}\\&=&\lambda\cdot\left(\lambda^{-1}\cdot\vec{v}\right)&&\phantom{xx}\color{blue}{\text{zero vector}}\\&=&\vec{v}&&\phantom{xx}\color{blue}{\text{associativity}}\\\end{array}\] Using the first rule, we conclude that #\lambda\cdot\vec{0}# is equal to the zero vector #\vec{0}#.

5. We prove this statement by contradiction. Assume that the condition holds and #\lambda\ne0#. We will arrive at a contradiction as follows: \[\begin{array}{rclll}\vec{u}&=&1\cdot\vec{u}&&\phantom{xx}\color{blue}{\text{scalar one}}\\&=&\left(\lambda^{-1}\cdot\lambda\right)\cdot\vec{u}&&\\ &=&\lambda^{-1}\cdot\left(\lambda\cdot\vec{u}\right)&&\phantom{xx}\color{blue}{\text{associativity}}\\ &=&\lambda^{-1}\cdot\left(\vec{0}\right)&&\phantom{xx}\color{blue}{\text{assumption}}\\ &=&\vec{0}&&\phantom{xx}\color{blue}{\text{part 4}}\end{array}\]

Below are examples of commonly used vector spaces.

The Euclidean plane #\mathbb{E}^2# and the Euclidean space #\mathbb{E}^3# are vector spaces.

The rules that define a vector space all hold for these examples, as discussed.

New example