Linear Algebra: Vector Spaces

Example Vector Spaces


Mathematical objects are often constructed in concrete forms to solve concrete problems before they are later abstracted and generalized by mathematicians. However, it can often be tedious to study mathematical objects if we do not first have a reason to - there are an infinite number of arbitrary mathematical objects one could construct - who's to say which are worth studying?. Concrete examples provide motivation to study vector spaces (and all abstract mathematical objects), and they also showcase the elegance of the abstracted model.

The best known vector spaces are the Euclidean spaces $\mathbb{R}^n$ and $\mathbb{C}^n$, although there are other less familiar examples as well.

In order to show that a set and an attendant pair of addition and scalar multiplication functions form a vector space, you must show that they satisfy all of the definitional requirements of a vector space.


Problems

  1. The symbol $\mathbb{R}$ denotes the set of all real numbers, and the symbol $\mathbb{R}^n$ denotes the $n$-dimensional Cartesian product of $\mathbb{R}$ with itself. In set builder notation, $\mathbb{R}^n = \left\{ \langle r_1, \ldots, r_n \rangle : r_1, \ldots, r_n \in \mathbb{R} \right\}$. Review the Cartesian products of sets if this is confusing.

    In plain English, $\mathbb{R}^n$ denotes all of the lists of length $n$ that are made up of real numbers. For example, $\langle -10.2, e\rangle$ is an element in $\mathbb{R}^2$, and $\langle 3, 2.7, 0\rangle$ and $\langle \pi, 14, 100 \rangle$ are both elements in $\mathbb{R}^3$. 

    Define the vector addition function, $+$, as $\langle a_1, \ldots, a_n \rangle + \langle b_1, \ldots, b_n \rangle = \langle a_1 + b_1, \ldots, a_n + b_n \rangle$, where $a_1, \ldots, a_n, b_1, \ldots, b_n \in \mathbb{R}$.

    Additionally, define the scalar multiplication function, $\cdot$, as $c \cdot \langle a_1, \ldots, a_n \rangle = \langle c \cdot a_1, \ldots, c \cdot a_n \rangle$ for all $c, a_1, \ldots, a_n \in \mathbb{R}$, where $\cdot$ within the vector denotes the standard multiplication of two real numbers.

    Show that $(\mathbb{R}^n, \mathbb{R}, +, \cdot)$ form a vector space.

    In order to show that something is a vector space, we must show that it meets all of the requirements in the definition of a vector space.

    First, let's show that the vector addition function meets the requirement that $u + v \in \mathbb{R}^n$ for all $u, v \in R^n$. By the given definition, $u + v = \langle u_1 + v_1, \ldots, u_n + v_n \rangle $. Since the sum of two real numbers is itself a real number, the list of numbers produced by the addition function is a member of $\mathbb{R}^n$, so the condition is satisfied.

    Let's now show that the scalar multiplication function checks out. Let $c \in \mathbb{R}$ and $v \in \mathbb{R}^n$. By the definition given, $c \cdot v = c \cdot \langle c \cdot v_1, \ldots, c \cdot v_n \rangle$. Since the produce of two real numbers is itself a real number, the list of numbers produced by the multiplication function is a member of $\mathbb{R}^n$, so the condition is satisfied.

    Next, let's show that the conditions on these functions are satisfied

    Commutativity of Addition:

    Let $u, v \in \mathbb{R}^n$. Then

    $u + v = \langle u_1 + v_1, \ldots, u_n + v_n \rangle$ Definition of vector addition
    $u + v = \langle v_1 + u_1, \ldots, v_n + u_n \rangle$ Commutativity of addition in $\mathbb{R}$
    $u + v = v + u \rangle$ Definition of vector addition

    Associativity of Addition:

    Let $u, v, w \in \mathbb{R}^n$. Then

    $u + (v + w) = u + \langle v_1 + w_1, \ldots, v_n + w_n \rangle$ Definition of vector addition
    $u + (v + w) = \langle u_1 + (v_1 + w_1), \ldots, u_n + (v_n + w_n) \rangle$ Definition of vector addition
    $u + (v + w) = \langle (u_1 + v_1) + w_1, \ldots, (u_n + v_n) + w_n \rangle$ Associativity of addition in $\mathbb{R}$
    $u + (v + w) = \langle u_1 + v_1, \ldots, u_n + v_n \rangle + w$ Definition of vector addition
    $u + (v + w) = (u + v) + w$ Definition of vector addition

    Additive Identity:

    Let $0$ denote the vector in $\mathbb{R}^n$ whose elements are all 0, and let $v \in \mathbb{R}^n$. Then

    $0 + v = \langle 0 + v_1, \ldots, 0 + v_n \rangle$ Definition of $0$ and of vector addition
    $0 + v = \langle v_1, \ldots, v_n \rangle$ Additive identity in $\mathbb{R}$ (i.e. $0 + r = r$ for all $r \in \mathbb{R}$.)
    $0 + v = v$ Definition of a vector in $\mathbb{R}^n$

    Multiplicative Identity:

    We know that $1$ is a real number is the multiplicative identity in the set of real numbers. Let $v \in \mathbb{R}^n$.

    $1 \cdot v = \langle 1 \dot v_1, \ldots, 1 \cdot v_n \rangle$ Definition of scalar multiplication
    $1 \cdot v = \langle v_1, \ldots, v_n \rangle$ Multiplicative identity in $\mathbb{R}$ (i.e. $1 \cdot r = r$ for all $r \in \mathbb{R}$.)
    $1 \cdot v = v$ Definition of a vector in $\mathbb{R}^n$

    Additive Inverse:

    Let $v \in \mathbb{R}^n$ and let $-v$ denote $\langle -v_1, \ldots, -v_n \rangle$. Since each $v_1, \ldots v_n$ are in $\mathbb{R}$, we know that $-v_1, \ldots, -v_n$ are also in $\mathbb{R}$, since each element of $\mathbb{R}$ has an additive inverse. As a result, $-v \in \mathbb{R}^n$. Then

    $v + (-v) = \langle v_1 + (-v_1), \ldots, v_n + (-v_n) \rangle$ Definition of vector addition
    $v + (-v) = \langle 0, 0, \ldots, 0 \rangle$ Additive inverse in $\mathbb{R}$ (i.e. $r + (-r) = 0$ for all $r \in \mathbb{R}$.)
    $v + (-v) = 0$ Definition of a vector in $\mathbb{R}^n$

    Distributive Law (Part 1):

    Let $a \in \mathbb{R}$ and $u, v \in \mathbb{R}^n$. Then

    $a \cdot (u + v) = a \cdot \langle u_1 + v_1, \ldots, u_n + v_n \rangle$ Definition of vector addition
    $a \cdot (u + v) = \langle a \cdot (u_1 + v_1), \ldots, a \cdot (u_n + v_n) \rangle$ Definition of scalar multiplication
    $a \cdot (u + v) = \langle a \cdot u_1 + a \cdot v_1, \ldots, a \cdot u_n + a \cdot v_n \rangle$ Distributive property in $\mathbb{R}^n$
    $a \cdot (u + v) = \langle a \cdot u_1, \ldots, a \cdot u_n \rangle + \langle a \cdot v_1, \ldots, a \cdot v_n \rangle$ Definition of vector addition
    $a \cdot (u + v) = a \cdot u + a \cdot v$ Definition of scalar multiplication

    Distributive Law (Part 2):

    Let $a, b \in \mathbb{R}$ and $v \in \mathbb{R}^n$. Then

    $(a + b) \cdot v = \langle (a + b) \cdot v_1, \ldots, (a + b) \cdot v_n \rangle$ Definition of scalar multiplication
    $(a + b) \cdot v = \langle a \cdot v_1 + b \cdot v_1, \ldots, a \cdot v_n + b \cdot v_n \rangle$ Distributive property in $\mathbb{R}^n$
    $(a + b) \cdot v = \langle a \cdot v_1, \ldots, a \cdot v_n \rangle + \langle b \cdot v_1, \ldots, b \cdot v_n \rangle$ Definition of vector addition
    $(a + b) \cdot v = a \cdot v + b \cdot v $ Definition of scalar multiplication
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