### 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}#

*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}#.

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#.

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#.

Below are examples of commonly used vector spaces.

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