Linear Algebra: Vector Spaces
Example Vector Spaces
In this section, we'll flesh out our abstract definition of a vector space with several concrete examples. We'll also go over common points of notation.
The best known vector spaces are the Euclidean spaces $\mathbb{R}^n$ and $\mathbb{C}^n$. Such vector spaces are referred to by just their "$V$" sets, without specifying the accompanying field and defining the vector addition and scalar multiplication functions. This is because these vector spaces are so commonly used that these other three components are assumed to be the standard, universally agreedupon definitions. In fact, for any field $F$, there is a standard definition for the vector space $F^n$, which you'll encounter in the problem set below. There are other standard definitions of other vector spaces, and for less common vector spaces, all four components will be explicitly defined.
Another point of notation is the usage of square brackets rather than parentheses when denoting vectors in $F^n$. For example, the ordered pair $(2, 7) \in \mathbb{R}^2$ is instead written as $[2, 7]$ to highlight the fact that we are dealing with vectors and the specific linear algebra context that they entail. Sometimes angle brackets are used instead, but we stick with square brackets, as they are more consistent with the matrix notation we'll encounter later.
In order to show that a set, a field, and an attendant pair of vector addition and scalar multiplication functions collectively form a vector space, you must show that they satisfy all of the definitional requirements of a vector space. It may help to pull up the definitions from the previous section in a separate tab or window and refer back to them as necessary.
Problems
A trivial vector space is a set containing just the $0$ vector, $\{0\}$, where $c0 = 0$ for all $c \in F$. A nontrivial vector space is any vector space that is not trivial.
Show that the trivial vector space is a vector space.
We check that $\{0\}$ fulfills all the requirements of a vector space:

Additive Identity: $0 \in \{0\}$ by definition, so $0 + 0 = 0$.

Closed Under Addition: $0 + 0 = 0 \in \{0\}$.

Associativity of Addition: $0 + (0 + 0) = 0 + 0 = (0 + 0) + 0$.

Commutativity of Addition: $0 + 0 = 0 + 0$.

Additive Inverse: $0 + 0 = 0$, so $0 = 0$.

Closure under Multiplication: $c0 = 0 \in \{0\}$ for all $c \in F$ by definition.

Distributivity: $a(0+0) = a0 = 0 = 0 + 0 = a0 + a0$. Likewise, $(a+b)0 = 0 = 0 + 0 = a0 + b0$.

Standard definition of $F^n$ as a vector space: Let $F$ be a field. The symbol $F^n$ denotes the $n$dimensional Cartesian product of $F$ with itself. In set builder notation, $F^n = \left\{ [f_1, \ldots, f_n] : f_1, \ldots, f_n \in F \right\}$. (Recall that we use square brackets rather than parentheses when considering elements of $F^n$ as vectors rather than merely points or ordered tuples.)
Define the vector addition function, $+$, as $[a_1, \ldots, a_n] + [b_1, \ldots, b_n] = [a_1 + b_1, \ldots, a_n + b_n]$, where $a_1, \ldots, a_n, b_1, \ldots, b_n \in F$.
Additionally, define the scalar multiplication function, $\cdot$, as $c \cdot [a_1, \ldots, a_n] = [c \cdot a_1, \ldots, c \cdot a_n]$ for all $c, a_1, \ldots, a_n \in F$, where $\cdot$ within the vector denotes the multiplication of elements of a field.
Show that $F^n$ forms a vector space over $F$ with the above definitions of vector addition and scalar multiplication.
Notation: Any mention of $\mathbb{R}^n$, $\mathbb{C}^n$, or any other $F^n$ as a vector space will imply the definition given here unless explicitly stated otherwise.
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:
Closure Under Addition:
By definition, $u + v = [u_1 + v_1, \ldots, u_n + v_n]$. Since $F$ is closed under addition, it follows that $u_i + v_i \in F$. Therefore $u + v \in F^n$.
Closure Under Scalar Multiplication:
By definition, $c \cdot v = [c \cdot v_1, \ldots, c \cdot v_n]$. Since $F$ is closed under multiplication, it follows that each $c \cdot v_i \in F$. Therefore $c \cdot v \in F^n$.
Commutativity of Addition:
Let $u, v \in F^n$. Then
$u + v = [u_1 + v_1, \ldots, u_n + v_n]$ Definition of vector addition $u + v = [v_1 + u_1, \ldots, v_n + u_n]$ Commutativity of addition in $F$ $u + v = v + u$ Definition of vector addition Associativity of Addition:
Let $u, v, w \in F^n$. Then
$u + (v + w) = u + [v_1 + w_1, \ldots, v_n + w_n]$ Definition of vector addition $u + (v + w) = [u_1 + (v_1 + w_1), \ldots, u_n + (v_n + w_n)]$ Definition of vector addition $u + (v + w) = [(u_1 + v_1) + w_1, \ldots, (u_n + v_n) + w_n]$ Associativity of addition in $F$ $u + (v + w) = [u_1 + v_1, \ldots, u_n + v_n]+ w$ Definition of vector addition $u + (v + w) = (u + v) + w$ Definition of vector addition Additive Identity:
Let $0$ denote the vector in $F^n$ whose elements are all $0 \in F$, and let $v \in F^n$. Then
$0 + v = [0 + v_1, \ldots, 0 + v_n]$ Definition of $0$ and of vector addition $0 + v = [v_1, \ldots, v_n]$ Additive identity in $F$ $0 + v = v$ Definition of a vector in $F^n$ Scalar Multiplicative Identity:
$1$ is the multiplicative identity in $F$. Let $v \in F^n$. Then
$1 \cdot v = [1 \cdot v_1, \ldots, 1 \cdot v_n]$ Definition of scalar multiplication $1 \cdot v = [v_1, \ldots, v_n]$ Multiplicative identity in $F$ $1 \cdot v = v$ Definition of a vector in $F^n$ Additive Inverse:
Let $v = [v_1, \ldots, v_n] \in F^n$ and let $v$ denote $[v_1, \ldots, v_n]$. Since each $v_1, \ldots v_n$ are in $F$, we know that $v_1, \ldots, v_n$ are also in $F$, since each element of $F$ has an additive inverse. As a result, $v \in F^n$. Then
$v + (v) = [v_1 + (v_1), \ldots, v_n + (v_n) ]$ Definition of vector addition $v + (v) = [0, 0, \ldots, 0]$ Additive inverse in $F$ $v + (v) = 0$ Definition of $0$ vector in $F^n$ Distributive Law (Part 1):
Let $a \in F$ and $u, v \in F^n$. Then
$a \cdot (u + v) = a \cdot [u_1 + v_1, \ldots, u_n + v_n]$ Definition of vector addition $a \cdot (u + v) = [a \cdot (u_1 + v_1), \ldots, a \cdot (u_n + v_n)]$ Definition of scalar multiplication $a \cdot (u + v) = [a \cdot u_1 + a \cdot v_1, \ldots, a \cdot u_n + a \cdot v_n]$ Distributive property in $F$ $a \cdot (u + v) = [a \cdot u_1, \ldots, a \cdot u_n] + [a \cdot v_1, \ldots, a \cdot v_n]$ Definition of vector addition $a \cdot (u + v) = a \cdot u + a \cdot v$ Definition of scalar multiplication Distributive Law (Part 2):
Let $a, b \in F$ and $v \in F^n$. Then
$(a + b) \cdot v = [(a + b) \cdot v_1, \ldots, (a + b) \cdot v_n]$ Definition of scalar multiplication $(a + b) \cdot v = [a \cdot v_1 + b \cdot v_1, \ldots, a \cdot v_n + b \cdot v_n]$ Distributive property in $F$ $(a + b) \cdot v = [a \cdot v_1, \ldots, a \cdot v_n]+ \langle b \cdot v_1, \ldots, b \cdot v_n]$ Definition of vector addition $(a + b) \cdot v = a \cdot v + b \cdot v $ Definition of scalar multiplication Determine whether the following sets form vector spaces over the other sets:

$\mathbb{R}^8$ over $\mathbb{R}$

$\mathbb{Q}^n$ over $\mathbb{Q}$

$\mathbb{C}$ over $\mathbb{C}$

$\mathbb{Z}^4$ over $\mathbb{Z}$
1, 2, and 3 are all vector spaces by the prior proof, as $\mathbb{R}$, $\mathbb{Q}$, and $\mathbb{C}$ are all fields. 4 is not a vector space, since $\mathbb{Z}$ is not a field.
