Vector space

We have already noticed some similarities between arrows in space and lists of numbers. But what inherently makes those objects similar? Why is it so natural to apply geometric reasoning to seemingly non-geometric objects, such as lists of numbers? Are there any other objects, which could be "geometrized" in a resembling manner?

Formal definition

You already know the definition of a vector and vector operations, so let's learn about collection of vectors. A vector space is a non-empty set $V$, elements of which are called vectors, with operations of addition, denoted by "$+$", and multiplication by any real number (scalar), denoted by "$\lambda \cdot$", where $\lambda$ is a real number. These operations must satisfy the following properties, called the axioms of vector space:

• Commutativity of vector addition.

$$p + q = q+p, \quad \text{for all } p,q\in V.$$

• Associativity of vector addition.

$$(p + q) + w = p + (q + w), \quad \text{for all } p,q,w \in V.$$

• Existence of a zero vector. In $V$ exists an element $\boldsymbol{0}$ such as:

$$0 \cdot p = \boldsymbol{0}, \quad \text{for all } p\in V.$$

• Existence of the opposite vector. For all vectors $p \in V$ exists a vector $-p$ such as:

$$p + (-p) = \boldsymbol{0}.$$

• Associativity of multiplication.

$$\lambda(\mu p) = (\lambda \mu) p, \quad \text{ for all } \lambda, \mu \in \mathbb{R},\, p \in V$$

• Distributivity of multiplication regarding scalar addition.

$$(\lambda + \mu)p = \lambda p + \mu p, \quad \text{ for all } \lambda, \mu \in \mathbb{R},\, p \in V$$

• Distributivity of multiplication regarding vector addition.

$$\lambda(p + q) = \lambda p + \lambda q, \quad \text{ for all } \lambda \in \mathbb{R},\, p, q \in V$$

• Unitarity.

$1 \cdot p = p, \quad \text{for all } p\in V$

Don't think that this realization was easy. It actually took centuries to come up with that definition. Abstract vector spaces were thoroughly researched only in the 20th century. Therefore, it is completely normal to feel a little confused by it at first. Give it time, you will feel much more confident as you continue to work with vector spaces. It is a rewarding experience. The rundown of this definition is that vectors are objects, that you can work with almost as easily as with arrows in space or real numbers, since the manipulations with them share common nature.

Consider the list of properties in the definition as a checklist. You can check whether the object you work with is a vector space. And if it is a vector space, then it automatically means, that the rich set of statements, that are true for all vector spaces, is true for the objects, you work with.

Examples of vector spaces

In the prerequisite topics, we discussed the lists of numbers and operations with them. Let's try to verify, whether the lists of numbers are vectors. For clarity of notation, we will check it with lists of two numbers, but it could be done with lists of any number of components.

Commutativity of vector addition:

$$\begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \begin{pmatrix}q_1 \\ q_2\end{pmatrix} = \begin{pmatrix}p_1 + q_1 \\ p_2 + q_2\end{pmatrix} = \begin{pmatrix}q_1 + p_1 \\ q_2 + p_2\end{pmatrix} = \begin{pmatrix}q_1 \\ q_2\end{pmatrix} + \begin{pmatrix}p_1 \\ p_2\end{pmatrix}.$$

Associativity of vector addition:

$$(\begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \begin{pmatrix}q_1 \\ q_2\end{pmatrix} ) + \begin{pmatrix}w_1 \\ w_2\end{pmatrix} = \begin{pmatrix}p_1 + q_ 1\\ p_2 + q_2 \end{pmatrix} +\begin{pmatrix}w_1 \\ w_2\end{pmatrix} = \begin{pmatrix}p_1 + q_ 1 + w_1 \\ p_2 + q_2 + w_2\end{pmatrix} = \begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \begin{pmatrix}q_1 + w_1\\ q_2 + w_2 \end{pmatrix} = \begin{pmatrix}p_1 \\ p_2\end{pmatrix} + (\begin{pmatrix}q_1 \\ q_2\end{pmatrix} + \begin{pmatrix}w_1 \\ w_2\end{pmatrix}).$$

Existence of a zero vector.

$$0 \cdot \begin{pmatrix}p_1 \\ p_2\end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix}.$$

Zero vector is $\begin{pmatrix}0 \\ 0\end{pmatrix}$.

Existence of the opposite vector. The opposite vector for $\begin{pmatrix}p_1 \\ p_2\end{pmatrix}$ is $\begin{pmatrix}-p_1 \\ -p_2\end{pmatrix}$:

$$\begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \begin{pmatrix}-p_1 \\ -p_2\end{pmatrix} = \begin{pmatrix}p_1 - p_1 \\ p_2 - p_2 \end{pmatrix} = \begin{pmatrix}0 \\ 0\end{pmatrix}$$

Associativity of multiplication:

$$\lambda(\mu \begin{pmatrix}p_1 \\ p_2\end{pmatrix} ) = \lambda\begin{pmatrix}\mu p_1 \\ \mu p_2\end{pmatrix} = \begin{pmatrix}\lambda\mu p_1 \\ \lambda\mu p_2\end{pmatrix} = (\lambda\mu) \begin{pmatrix}p_1 \\ p_2\end{pmatrix}.$$

Distributivity of multiplication regarding scalar addition:

$$(\lambda + \mu) \begin{pmatrix}p_1 \\ p_2\end{pmatrix} = \begin{pmatrix}(\lambda + \mu) p_1 \\ (\lambda + \mu) p_2\end{pmatrix} = \begin{pmatrix}\lambda p_1 + \mu p_1 \\ \lambda p_2 + \mu p_2 \end{pmatrix} = \begin{pmatrix}\lambda p_1 \\ \lambda p_2\end{pmatrix} + \begin{pmatrix}\mu p_1 \\ \mu p_2\end{pmatrix} = \lambda \begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \mu \begin{pmatrix}p_1 \\ p_2\end{pmatrix}.$$

Distributivity of multiplication regarding vector addition:

$$\lambda(\begin{pmatrix}p_1 \\ p_2\end{pmatrix} + \begin{pmatrix}q_1 \\ q_2\end{pmatrix}) = \lambda \begin{pmatrix}p_1 + q_1 \\ p_2 + q_2\end{pmatrix} = \begin{pmatrix}\lambda(p_1 + q_1) \\ \lambda (p_2 + q_2)\end{pmatrix} = \begin{pmatrix}\lambda p_1 + \lambda q_1 \\ \lambda p_2 + \lambda q_2\end{pmatrix}.$$

Unitarity.

$$1 \cdot \begin{pmatrix}p_1 \\ p_2\end{pmatrix} = \begin{pmatrix}1 \cdot p_1 \\ 1 \cdot p_2\end{pmatrix} = \begin{pmatrix}p_1 \\ p_2\end{pmatrix} .$$

All the axioms of vector space are true for lists of numbers with "intuitively" defined operations of addition and multiplication of a number.

Another example of a vector space is a set of all polynomials of a degree less than or equal to $2$. Let's denote it as $P$. Does $P$ satisfy all the axioms of vector space? Let's check!

1. Commutativity of addition: For all $p(x), q(x)$ in $P$, $p(x) + q(x) = q(x) + p(x)$. This is a property of polynomial addition.
2. Associativity of addition: For all $p(x), q(x), r(x)$ in $P$, $(p(x) + q(x)) + r(x) = p(x) + (q(x) + r(x))$. This is also a property of polynomial addition.
3. Existence of a zero vector: There exists a zero polynomial, denoted as $0$, such that for all $p(x)$ in $P$, $p(x) + 0 = p(x)$. This zero polynomial is of degree less than or equal to $2$, so it is in $P$.
4. Existence of the opposite vector: For every $p(x)$ in $P$, there exists a polynomial $-p(x)$ in $P$ such that $p(x) + (-p(x)) = 0$. The polynomial $-p(x)$ is of degree less than or equal to $2$, so it is in $P$.
5. Associativity of multiplication: For all $p(x)$ in $P$ and scalars $c$ and $d$, $(c\cdot d)\cdot p(x) = c\cdot(d\cdot p(x))$. This is a property of scalar multiplication.
6. Distributivity of multiplication regarding scalar addition: Also, for all $p(x)$ in $P$ and scalars $c$ and $d$, $(c + d)\cdot p(x) = c\cdot p(x) + d\cdot p(x)$. This is also a property of scalar multiplication and polynomial addition.
7. Distributivity of multiplication regarding vector addition: For all $p(x), q(x)$ in $P$ and scalar $c$, ⁣$c\cdot(p(x) + q(x)) = c\cdot p(x) + c\cdot q(x)$. This is once more a property of scalar multiplication and polynomial addition.
8. Unitarity: For every $p(x)$ in $P$: $1\cdot p(x) = p(x)$ according to the definition of scalar multiplication of a polynomial.

Since all eight axioms are satisfied, we can conclude that the set of all polynomials of degree less than or equal to 2 forms a vector space.

Example of proof in abstract vector space

We discussed that the notion of abstract vector space is needed to prove properties, that are common for all vector spaces. Axioms of vector space require from zero vector to be the result of multiplication of any vector by zero. They do not clearly state the connection between the operation of addition and zero vector. Let's try to prove that in any vector space exists only one vector $\tilde{\boldsymbol{0}}$ such as

$$p + \tilde{\boldsymbol{0}} = p, \quad \text{for all }p\in V$$

and $\tilde{\boldsymbol{0}} = \boldsymbol{0}$.

First, let's prove that

$$p + \boldsymbol{0} = p, \quad \text{for all }p\in V.$$

It is easy to prove through straightforward application of axioms

$$p + \boldsymbol{0} \stackrel{\text{zero vec. def.}}{=} p + 0\cdot p \stackrel{unitarity }{=} 1\cdot p + 0\cdot p \stackrel{distributivity}{=} (1 + 0) \cdot p = 1 \cdot{p} \stackrel{unitarity }{=} p.$$

Now, let's assume that there exists some other $\tilde{\boldsymbol{0}}$ such as

$$p + \tilde{\boldsymbol{0}} = p, \quad \text{for all }p\in V.$$

Then

$$\boldsymbol{0}+\tilde{\boldsymbol{0}} = \boldsymbol{0},\\ \tilde{\boldsymbol{0}}+\boldsymbol{0} = \tilde{\boldsymbol{0}}.$$

But through associativity, we can get that

$$\boldsymbol{0}+\tilde{\boldsymbol{0}} = \tilde{\boldsymbol{0}}+\boldsymbol{0}.$$

Those three equations combined yield $\boldsymbol{0}= \tilde{\boldsymbol{0}}$. The proof is complete. We proved that for all possible vector spaces exists only one vector, which doesn't add anything to any vector, and this vector is a zero vector, defined in the axioms.

The practice problems of this topic can be answered through similar simple proofs.

Conclusion

In this topic, you got acquainted with the definition of an abstract vector space. Here are several crucial points regarding vector spaces and their definitions:

• A vector space is a non-empty set $V$, elements of which are called vectors, with operations of addition and multiplication by any real number (scalar), which satisfy these axioms: commutativity of vector addition, associativity of vector addition and multiplication, the existence of opposite and zero vectors, distributivity of multiplication regarding scalar and vector addition, and unitarity.
• In order to check whether a set of objects is a vector space, you should check if all the statements that are true for any vector space are true for all objects of the set.
• A list of numbers and a set of all polynomials of degree less than or equal to $2$ are two basic examples of vector spaces.