Vector space – Serlo

In this chapter we define the concept of vector space, which is the basis for linear algebra. Vector spaces consist of objects that can be added and scaled. The addition and the scaling have certain properties.

Recap: Intuition behind a vector space

→ Main article: Introduction: Vector space

The notion of a vector space is an abstract generalization of the plane $\mathbb {R} ^{2}$ and the space $\mathbb {R} ^{3}$. In these two sets we can think of vectors as arrows that can be added and scaled:

• 
  Addition of two vectors in $\mathbb {R} ^{2}$
• 
  scalar multiplication in $\mathbb {R} ^{2}$

This is where the general vector space notion comes in: There are more sets with similar properties. Also polynomials can be added and scaled. These relations are very similar to those of vectors. So the addition and scalar multiplication of polynomials of second degree corresponds with the respective operations in $\mathbb {R} ^{3}$:

${\begin{array}{ccc}{\color {Red}7}x^{2}&+{\color {OliveGreen}3}x&+({\color {NavyBlue}-2})&\leftrightarrow &({\color {Red}7},{\color {OliveGreen}3},{\color {NavyBlue}-2})^{T}\\&+&&&+\\{\color {Red}1}x^{2}&+{\color {OliveGreen}0}x&+{\color {NavyBlue}6}&\leftrightarrow &({\color {Red}1},{\color {OliveGreen}0},{\color {NavyBlue}6})^{T}\\&=&&&=\\{\color {Red}8}x^{2}&+{\color {OliveGreen}3}x&+{\color {NavyBlue}4}&\leftrightarrow &({\color {Red}8},{\color {OliveGreen}3},{\color {NavyBlue}4})^{T}\end{array}}$

and:

${\begin{array}{ccc}&-1&&&-1\\&\cdot &&&\cdot \\{\color {Red}7}x^{2}&+{\color {OliveGreen}3}x&+({\color {NavyBlue}-2})&\leftrightarrow &({\color {Red}7},{\color {OliveGreen}3},{\color {NavyBlue}-2})^{T}\\&=&&&=\\{\color {Red}-7}x^{2}&+({\color {OliveGreen}-3})x&+{\color {NavyBlue}2}&\leftrightarrow &({\color {Red}-7},{\color {OliveGreen}-3},{\color {NavyBlue}2})^{T}\\\end{array}}$

The vector space notion hence generalizes the plane $\mathbb {R} ^{2}$ and the space $\mathbb {R} ^{3}$ to further structures with similar properties. Vectors are objects which can be added and scaled like vectors from $\mathbb {R} ^{2}$ or $\mathbb {R} ^{3}$.

Definition of a vector space

We want to define a general vector space $V$ over the field $K$. For this we use the two operations $\boxplus$ for the vector addition and $\boxdot$ for the vector scaling.

Definition (vector space)

Let $V$ be a non-empty set with an inner operation $\boxplus :V\times V\to V$ (of vector addition) and an outer operation $\boxdot :K\times V\to V$ (of scalar multiplication). The set $V$ with these two operations is called vector space over the field $K$ or alternatively $K$-vector space if the following axioms hold:

• $V$ together with the operation $\boxplus$ forms an abelian group (missing). That is, the following axioms are satisfied:
  1. associative law: For all $v,w,z\in V$ we have that: $v\boxplus (w\boxplus z)=(v\boxplus w)\boxplus z$.
  2. commutative law: For all $v,w\in V$ we have that: $v\boxplus w=w\boxplus v$.
  3. Existence of a neutral element: There is an element $0\in V$ such that for all $v\in V$ we have that: $v\boxplus 0=v$. This vector $0$ is called neutral element of addition or zero vector.
  4. Existence of an inverse element: To every $v\in V$ there exists an element $y\in V$ such that we have $v\boxplus y=0$. The element $y$ is called inverse element to $v$. Instead of $y$ we also write $-v$.
• In addition, the following axioms of scalar multiplication $\boxdot$ must be satisfied:
  1. Scalar distributive law: For all $\lambda ,\rho \in K$ and all $v\in V$ we have that: $(\lambda +\rho )\boxdot v=(\lambda \boxdot v)\boxplus (\rho \boxdot v)$.
  2. Vector distributive law: For all $\lambda \in K$ and all $v,w\in V$ we have that: $\lambda \boxdot (v\boxplus w)=(\lambda \boxdot v)\boxplus (\lambda \boxdot w)$
  3. Associative law for scalars: For all $\lambda ,\rho \in K$ and all $v\in V$ it holds that: $(\lambda \cdot \rho )\boxdot v=\lambda \boxdot (\rho \boxdot v)$.
  4. Neutral element of scalar multiplication: For all $v\in V$ and for $1\in K$ (the neutral element of multiplication in $K$) we have that: $1\boxdot v=v$. The 1 is called neutral element of scalar multiplication.

Instead of "$V$" one often writes "$(V,\boxplus ,\boxdot )$". The last notation makes clear that the set $V$ includes the operations $\boxplus$ and $\boxdot$.

Hint

We use the symbols "$\boxplus$" and "$\boxdot$" to distinguish them from addition "$+$" and multiplication "$\cdot$". In the literature this distinction is often not made and from the context it becomes clear whether for example "$+$" means an addition of numbers or of vectors.

Remarks concerning the Definition

The scalar and the vector distributive law of scalar multiplication differ in that one statement is made about the addition in the field and one about the addition in the vector space. Thus the scalar distributive law establishes a relation between the field addition and the vector addition:

${\color {OliveGreen}\underbrace {(\lambda +\rho )} _{\text{field addition}}}\boxdot v={\color {Blue}\underbrace {(\lambda \boxdot v)\boxplus (\rho \boxdot v)} _{\text{vector addition}}}$

By contrast, the vector distributive law says something about how vector addition behaves under a scaling:

$\lambda \boxdot {\color {Blue}\underbrace {(v\boxplus w)} _{\text{vector addition}}}={\color {Blue}\underbrace {(\lambda \boxdot v)\boxplus (\lambda \boxdot w)} _{\text{vector addition}}}$

In the associative law for scalars, multiplication in the field and scalar multiplication are applied once to the left of the equality sign, while scalar multiplication is applied once to the right of the equality sign:

${\color {Blue}\underbrace {{\color {OliveGreen}\underbrace {(\lambda \cdot \rho )} _{\text{field multiplication}}}\boxdot v} _{\text{scaling}}}={\color {Blue}\underbrace {\lambda \boxdot {\color {Blue}\underbrace {(\rho \boxdot v)} _{\text{scaling}}}} _{\text{scaling}}}$

The scalar distributive law $(\lambda +\rho )\boxdot v=(\lambda \boxdot v)\boxplus (\rho \boxdot v)$ behaves exactly like the distributive law in fields. So in the future, instead of awkwardly writing $(\lambda +\rho )\boxdot v=(\lambda \boxdot v)\boxplus (\rho \boxdot v)$, we will use the easier notion $(\lambda +\rho )\cdot v=(\lambda \cdot v)+(\rho \cdot v)$. So we do not distinguish in our notation between field addition and vector addition and also not between field multiplication and scalar multiplication. Which operation is meant, results in each case from the context.

This holds analogously for the vectorial distributive law as well as for the associative law for scalars. With the vectorial distributive law, the equation $\lambda \boxdot (v\boxplus w)=(\lambda \boxdot v)\boxplus (\lambda \boxdot w)$ becomes $\lambda \cdot (v+w)=(\lambda \cdot v)+(\lambda \cdot w)$. With the associative law for scalars, $(\lambda \cdot \rho )\boxdot v=\lambda \boxdot (\rho \boxdot v)$ becomes the expression $(\lambda \cdot \rho )\cdot v=\lambda \cdot (\rho \cdot v)$. The analogy of the individual vector space axioms to the respective field axioms thus justifies that we can also use the symbols "$+$" and "$\cdot$" for vector spaces.

Vectors point into directions

How can we visualize vectors? First, we take a look at $\mathbb {R} ^{3}$, where such a notion is intuitive, and then generalize it.

Intuition: vectors in $\mathbb {R} ^{3}$

Vectors $v\neq 0$ in $\mathbb {R} ^{3}$ can be visualized by arrows starting at the origin and pointing in certain directions. We can think of the direction of a vector $v$ as the straight line in space containing all multiples of the vector. Mathematically, we can specify this straight line by $\{\lambda \cdot v|\lambda \in \mathbb {R} \}$. However, we see that several vectors can then have the same direction line. For instance, the different vectors $v,{\tfrac {1}{2}}v$ and $3v$ point in the same direction, so they have the same direction line. Thus vectors are characterized by more than their direction. They also have a length ratio to each other. For example, ${\tfrac {1}{2}}v$ has a ratio of ${\tfrac {1}{2}}:1$ to $v$, since stretching the vector $v$ by ${\tfrac {1}{2}}$ yields the vector ${\tfrac {1}{2}}v$ which is exactly half as long.

So there are two kinds of information contained in a $\mathbb {R} ^{3}$ vector: First, in which direction the vector points, and second,in which relation its length stands to that of other vectors pointing in the same direction. Both information together determine unambiguously where the vector points.

Directions and ratios in a more subtle vector space

That the notion of direction and the notion of width are meaningful in $\mathbb {R} ^{n}$ is obvious to us. Now we try to transfer the notion of vectors to other, more complicated vector spaces. To do this, we look at the vector space $(\mathbb {Z} /5\mathbb {Z} )^{2}$ over the field with five elements $\mathbb {Z} /5\mathbb {Z}$. One can show that $\mathbb {Z} /5\mathbb {Z}$ is a field. .

We can think of this field as a circle, which has a clock face. But unlike a clock, it has not twelve but only five digits representing the five elements of the field $\mathbb {Z} /5\mathbb {Z}$:

The field Z modulo 5Z

$\mathbb {Z} /5\mathbb {Z}$ consists of the five numbers $\{0,1,2,3,4\}$, where these have a cyclic structure:

$\ldots \,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,4\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,0\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,1\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,2\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,3\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,4\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,0\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,1\,{\color {OliveGreen}{\stackrel {+1}{\longrightarrow }}}\,\ldots$

On this circle you can calculate similarly as on a clock. For instance, $3+4=2$:

Addition in Z modulo 5Z

Now we consider what the vector space $(\mathbb {Z} /5\mathbb {Z} )^{2}$ must look like based on this idea. Since we need two independent dimensions, we imagine $(\mathbb {Z} /5\mathbb {Z} )^{2}$ as two independent circles. We take one circle and add the other circle at the origin. Altogether we get a torus, which is provided with a grid because of the division of the circles with digits. Only the grid points on the torus are elements of the vector space $(\mathbb {Z} /5\mathbb {Z} )^{2}$:

The vector space Z/5Z^2

The axes of $\mathbb {R} ^{2}$ in this example of vector space $(\mathbb {Z} /5\mathbb {Z} )^{2}$ correspond to the two circles forming the torus. Also in this example vectors have directions. If we fix a vector $v$, it points in a concrete direction. All multiples of the vector form the directional vectors $\lambda v:\lambda \in \mathbb {Z} /5\mathbb {Z} \}$. Ratios are also expressible. Since $\mathbb {Z} /5\mathbb {Z}$ consists only of the five numbers $\{0,1,2,3,4\}$, also only these five ratios are possible. For instance, $(3,1)^{T}$ is the double of $(4,3)^{T}$, because:

$2\cdot {\begin{pmatrix}4\\3\end{pmatrix}}={\begin{pmatrix}2\cdot 4\\2\cdot 3\end{pmatrix}}={\begin{pmatrix}3\\1\end{pmatrix}}$

Generalization of directions and ratios

In the previous considerations we have seen that vectors point in directions. Different vectors of the same direction have a concrete ratio, which corresponds to an element of the underlying field. So let us take a general vector space $V$ over the field $K$. For a direction vector $v\in V$ we can define the corresponding direction as $\lambda v:\lambda \in K\}$. All vectors of these straight lines have a concrete relation to the direction vector $v$, which is defined by the scalar $\lambda \in K$.