This article is an introduction to vectors and subspaces. We use the terms vector space and linear space as synonyms.

## Introduction to Vectors

Very precisely, here is what we mean by vector space.

**Definition**. Let $\mathbb{F}$ be a field (whose elements are called **scalars**) and let $\mathbb{V}$ be a nonempty set (whose elements are called **vectors**) on which two operations, called **addition** and **scalar multiplication**, have been defined. The addition operation (denoted by $+$), assigns to each pair $(u,v)\in \mathbb{V}\times \mathbb{V}$, a unique vector $u+v$ in $\mathbb{V}.$ The scalar multiplication operation (denoted by juxtaposition), assigns to each pair $(a,v)\in \mathbb{F}\times \mathbb{V}$ a unique vector $a v$ in $\mathbb{V}.$ We call $\mathbb{V}$ a **linear space** if the following axioms (A1)-(A8) are also satisfied.

(1) For all $u, v\in \mathbb{V}$, $u+v=v+u.$

(2) For all $u, v, w \in \mathbb{V}$, $(u+v)+w=u+(v+w).$

(3) There exists ${0}\in \mathbb{V}$ such that $v+{0}=v$ for all $v\in \mathbb{V}.$

(4) For every $v\in \mathbb{V}$, there exists $w\in \mathbb{V}$ such that $v+w={0}.$

(5) For all $v\in \mathbb{V}$, $1 v=v.$

(6) For all $a, b\in \mathbb{F}$ and $u\in \mathbb{V}$, $(a b) v=a (b v).$

(7) For all $a \in \mathbb{F}$ and $u, v\in \mathbb{V}$, $a(u+v)=a u+av.$

(8) For all $a, b \in \mathbb{F}$ and $u\in \mathbb{V}$, $(a+b)u=a u+ b u.$

If $\mathbb{F}=\mathbb{R}$ then $\mathbb{V}$ is a called a **real linear space**. If $\mathbb{F}=\mathbb{C}$ then $\mathbb{V}$ is called a **complex linear space**. We denote the zero vector as $0$, to distinguish between the zero vector and the zero $0$ in the field of scalars. For a short history.

## Immediate Theorems when Introducing Vectors

The abstract definition above allows us to take a formal introduction to vectors that is very productive.

**Theorem**. Every linear space $\mathbb{V}$ has a unique additive identity (denoted by $0$).

**Proof**. Let $u_1$ and $u_2$ be additive identities in $V$, then $v+u_1=v$ and $v+u_2=v$ for every $v\in V.$ Thus, $u_1=u_1+u_2=u_2+u_1=u_2$ as desired.

In our introduction, we are emphasizing the importance of the special elements in any vector space: the additive identity.

**Theorem**. Every $v\in V$ has a unique additive inverse, (denoted by $-v$).

**Proof**. Let $v_1$ and $v_2$ be additive inverses of $w$ in $V$, then $w+v_1=\mathbf{0}$ and $w+v_2=\mathbf{0}.$ Thus,

\begin{align} v_1 & = v_1+\mathbf{0} = v_1+(w+v_2) = (v_1+w)+v_2 =(w+v_1)+v_2\\ & =\mathbf{0}+v_2=v_2 \end{align} as desired.

**Theorem**. If $v\in V$, then $0\, v={0}.$

**Proof**. Let $v\in V$, then $v=1 v=(1+0) v= 1 v+0 v= v+0v$ which shows that $0 v$ is the additive identity of $V$, namely $0 v=\mathbf{0}.$

**Theorem**. If $a\in \mathbb{F}$, then $a\, 0=0.$

**Proof**. Let $a\in \mathbb{F}$, then $ a \mathbf{0}=a(\mathbf{0}+\mathbf{0})=a\mathbf{0}+a\mathbf{0} $ which shows that $a \mathbf{0}$ is the additive identity of $V$, namely $a \mathbf{0}=\mathbf{0}.$

**Theorem**. If $v\in V$, then $-(-v)=v.$

**Proof**. Let $v\in V$, then $ v+(-1)v=1 v+(-1) v=(1+(-1)) v=0 v= \mathbf{0} $ which shows that $(-1)v$ is the unique additive inverse of $v$ namely, $(-1)v=-v.$

**Theorem**. If $v\in V$, then $(-1)\, v=-v.$

**Proof**. Since $-v$ is the unique additive inverse of $v$, $v+(-v)=\mathbf{0}.$ Then $(-v)+v=\mathbf{0}$ shows that $v$ is the unique additive inverse of $-v$, namely, $v=-(-v)$ as desired.

**Theorem**. If $a\,v=0$, then $a=0$ or $v=0.$

**Proof**. Suppose $a\neq 0.$ If $a v =\mathbf{0}$ then $v=1 v=(a^{-1} a) v=a^{-1} (a v)=a^{-1} \mathbf{0}=\mathbf{0}.$ Otherwise $a=0$ as desired.

Let $\mathbb{V}$ be a linear space and $U$ a nonempty subset of $\mathbb{V}.$ If $U$ is a linear space with respect to the operations on $\mathbb{V}$, then $U$ is called a subspace of $\mathbb{V}.$ To fully understand this introduction to vectors, you will need to master the idea of a subspace.

**Theorem**. A subset $U$ of $\mathbb{V}$ is a linear subspace of $\mathbb{V}$ if and only if $U$ has the following properties:

(1) $U$ contains the zero vector of $\mathbb{V}$,

(2) $U$ is closed under the addition defined on $\mathbb{V}$, and

(3) $U$ is closed under the scalar multiplication defined on $\mathbb{V}.$

More generally, a subset $U$ of $\mathbb{V}$ is called a subspace of $\mathbb{V}$ if $U$ is also a vector space using the same addition and scalar multiplication as on $\mathbb{V}.$ Any vector space is a subspace of itself. The set containing just the $0$ vector is also a subspace of any vector space. Given any vector space with a nonzero vector $v$, the scalar multiples of $v$ is a vector subspace of $\mathbb{V}$ and is denoted by $\langle v \rangle.$ Because any linear space $\mathbb{V}$ has $\mathbb{V}$ and $0$ as subspaces, these subspaces are called the trivial subspaces of $\mathbb{V}.$ All other subspaces are called proper subspaces of $\mathbb{V}.$

**Example**. Give an example of a real linear space $\mathbb{V}$ and a nonempty set $S$ of $\mathbb{V}$ such that, whenever $u$ and $v$ are in $S$, $u+v$ is in $S$ but $S$ is not a subspace of $\mathbb{V}.$

**Example**. Give an example of a real linear space $\mathbb{V}$ and a nonempty set $S$ of $\mathbb{V}$ such that, whenever $u$ and $v$ are in $S$, $c u$ is in $S$ for every scalar $c$ but $S$ is not a subspace of $\mathbb{V}.$

**Example**. Show that $P_n[0,1]$ is a proper subspace of $C[0,1].$

**Example**. Show that $C'[0,1]$ (continuous first derivative) is a proper subspace of $C[0,1].$

**Example**. Show that $R[0,1]$ (Riemann integrable) is a proper subspace of $C[0,1].$

**Example**. Show that $D[0,1]$ (Differentiable functions) is a proper subspace of $C[0,1].$

## Linear Combinations

In any good introduction to vectors is the concept of linear combinations.

**Definition**. A ** linear combination** of a list of vectors $(v_1, v_2, \ldots, v_m)$ in $\mathbb{V}$ is a vector of the form $a_1 v_1 + a_2 v_2 + \cdots + a_m v_m $ where $a_1, a_2, \ldots, a_m \in k.$

**Theorem**. Let $U$ be a nonempty subset of a vector space $\mathbb{V}.$ Then $U$ is a subspace of $\mathbb{V}$ if and only if every linear combination of vectors in $U$ is also in $U.$

**Proof**. If $U$ is a subspace of $\mathbb{V}$, then $U$ is a vector space and so is closed under linear combinations by definition of vector space. Conversely, suppose every linear combination of vectors in $U$ is also in $U.$ Thus for any $a, b \in k$, $a u+b v \in U$ for every $u, v\in U.$ In particular, when $a=b=1$ then $u+v \in U$ and so $U$ is closed with respect to addition. Notice when $b=0$ and $a=-1$, then $-u\in U$ for every $u\in U$ and so $U$ is closed under inverses. Notice when $u=v$, $a=1$, and $b=-1$ then $u+(-u)=0\in U$ so $U$ contains the identity element. The rest of the axioms in the definition of a vector space hold by containment.

**Definition**. The intersection and union of subspaces is just the intersection and union of the subspaces as sets. The sum of subspaces $U_1, U_2, \ldots, U_m$ of a vector space $U$ is defined by $$ U_1+ U_2+\cdots +U_m = \{ u_1 + u_2+\cdots + u_m \mid u_i \in U_i \text{ for } 1\leq i \leq m \}. $$

## Unions and Intersections of Subspaces

Now that we have a good introduction to vectors going, we can discuss unions and intersections of subspaces. When do these operations on subspaces generate new subspaces?

**Theorem**. Let $\mathbb{V}$ be a linear space over a field $k.$ The intersection of any collection of subspaces of $\mathbb{V}$ is a subspace of $\mathbb{V}.$

**Proof**. Let $\{U_i\, |\, i \in I\}$ be a collection of subspaces where $I$ is some indexed set. Let $a,b\in k$ and $u,v\in \cap_{i\in I} U_i.$ Since each $U_i$ is a subspace of $\mathbb{V}$, $a u +b v\in U_i$ for every $i\in I.$ Thus $a u+b v\in \cap_{i\in I} U_i$ and therefore $\cap_{i\in I} U_i$ is a subspace of $\mathbb{V}.$

For example, the $x$-axis and the $y$-axis are subspaces on $\mathbb{R}^2$, yet the union of these axis is not.

**Theorem**. Let $\mathbb{V}$ be a linear space over a field $k.$ The union of two subspaces of $\mathbb{V}$ is a subspace of $\mathbb{V}$ if and only if one of the subspaces is contained in the other.

**Proof**. Suppose $U$ and $W$ are subspaces of $\mathbb{V}$ with $U\subseteq W.$ Then $U\cup W=W$ and so $U\cup W$ is also a subspace of $\mathbb{V}.$ Conversely, suppose $U$, $W$, $U\cup W$ are subspaces of $\mathbb{V}$ and suppose $u\in U.$ If $u\in W$ then $U$ is contained in $W$ as desired. Thus we assume, $u\not\in W.$ If $w\in W$, then $u+w \in U\cup W$ and either $u+w\in U$ or $u+w\in W.$ Notice $u+w\in W$ and $w\in W$ together yield $u\in W$ which is a contradiction. Thus $u+w\in U$ and so $w\in U$ which yields $W\subseteq U$ as desired.

**Theorem**. Let $\mathbb{V}$ be a linear space over a field $k.$ The sum $U_{1}+ U_2+\cdots +U_{m}$ is the smallest subspace containing each of the subspaces $U_1, U_2, \ldots, U_m.$

**Proof**. The sum of two subspaces is a subspace since the sum of two subspaces is closed under linear combinations. Thus $U_1, U_2, \ldots, U_m$ is a subspace containing $U_i$ for each $1\leq i \leq m.$ Let $U$ be another subspace containing $U_{i}$ for each $1\leq i \leq m.$ If $u\in U_{1}+ \cdots +U_{m}$, then $u$ has the form $u=u_1+\cdots + u_m$ where each $u_i\in U_i\subseteq U.$ Since $U$ is a subspace $u\in U$ and so $U_{1}+U_{2}+ \cdots +U_{m}$ is the smallest such subspace.

**Definition**. If $\{v_1, \ldots, v_m\}$ is a subset of a linear space $\mathbb{V}$, then the subspace of all linear combinations of these vectors is called the subspace generated (spanned) by $\{v_1, \ldots, v_m\}.$ The spanning set of the list of vectors $(\{v_1, \ldots, v_m\})$ in $\mathbb{V}$ is denoted by $$\mathop{span}(\{v_1, \ldots, v_m\}) = \{a_1 v_1 + a_2 v_2 + \cdots + a_m v_m \mid a_1, a_2, \ldots, a_m\in \mathbb{F} \}.$$

**Theorem**. The span of a list of vectors in $\mathbb{V}$ is the smallest subspace of $\mathbb{V}$ containing all the vectors in the list.

**Proof**. Let $(v_1, v_2, \ldots, v_n)$ be a list of vectors in $\mathbb{V}$ and let $S$ denote $\mathop{span}(v_1, v_2, \ldots, v_n).$ Clearly, $S$ contains $v_i$ for each $1\leq i \leq n.$ Let $u,v \in S$ and $a,b\in k.$ Then there exists $a_1, a_2, \ldots, a_n$ in $k$ and $b_1, b_2, \ldots, b_n$ in $k$ such that $u=a_1 v_1+\cdots a_n v_n$ and $v=b_1 v_1+ \cdots b_n v_n.$ Then $$a u+b v=(a a_1 +b b_1) v_1+ \cdots +(a a_n+b b_n) v_n$$ which shows $a u+b v\in S$ since $a a_i+b b_i \in k$ for each $1\leq i \leq n.$ Thus $S$ is a subspace containing each of the $v_i.$ Let $T$ be a subspace containing $v_i$ for $1 \leq i \leq n.$ If $s\in S$, then there exists $c_1, c_2, \ldots, c_n \in k$ such that $s=c_1 v_1+\cdots + c_n v_n.$ Since $v_i\in T$ for each $i$ and $T$ is closed under linear combinations (since $T$ is a subspace), $s\in T.$ Meaning $S \subseteq T$, so indeed $S$ is the smallest subspace of $\mathbb{V}$ containing all the vectors $v_i.$

**Definition**. Let $\emptyset$ denote the empty set. Then $\mathop{span}(\emptyset)={0}.$

## Exercises on Introduction to Vectors

**Example**. Let $A=\begin{matrix}1 & 1 \ 0 & 0 \end{matrix}.$ Show that $S=\{X\in M_{2\times 2} \mid AX=XA\}$ is a subspace of $M_{2\times 2}$ under the standard operations.

**Example**. Let $f_1=x^2+1, f_2=3x-1, f_3=2.$ Determine the subspace generated by $f_1, f_2, f_3$ in $P_4.$

**Example**. Let $A_1=\begin{bmatrix} 1 & 0 & 0 & 3 \\ 0 & 0 & 2 & 0 \end{bmatrix} $ and $A_2= \begin{bmatrix} 0 & 2 & 0 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix}.$ Determine the subspace generated by $A_1$ and $A_2$ in $R_{2\times 4}.$

**Example**. Describe $\mathop{span}(0).$

**Example**. Consider the subset $$S=\{x^3-2x^2+x-3, 2x^3-3x^2+2x+5, 4x^3-7x^2+4x-1, 4x^2+x-3\}$$ of $P.$ Show that $3x^3-8x^2+2x+16$ is in $\mathop{span} (S)$ by expressing it as a linear combination of the elements of $S.$

**Example**. Determine if the matrices $\begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}, \begin{bmatrix}-4 & 2 \\ 3 & 0 \end{bmatrix}, \begin{bmatrix} -1 & 0 \\ 2 & 1 \end{bmatrix}, \begin{bmatrix}0 & 0 \\ 0 & 3\end{bmatrix}$ span $M_{2\times 2}.$

## Exercises on Introduction to Vectors

**Exercise**. Show that the set of solution to a homogenous system form a linear space with standard operations.

**Exercise**. Show that the set of vectors for which a particular linear system has a solution is a linear space.

**Exercise**. Let $V=\{(x,y)\mid y=mx\}$, where $m$ is a fixed real number and $x$ is an arbitrary real number. Show that $\mathbb{V}$ is a linear space.

**Exercise**. Let $V=\{(x,y,x)\mid ax+by+cz=0\}$ where $a, b$ and $c$ are fixed real numbers. Show that $\mathbb{V}$ is a linear space with the standard operations.

**Exercise**. (** Matrix Space**) Show that the set $M_{m\times n}$ of all $m\times n$ matrices, with ordinary addition of matrices and scalar multiplication, forms a linear space.

**Exercise**. (** Polynomial Space**) Show that the set $P(t)$ of all polynomials with real coefficients, under the ordinary operations of addition of polynomials and multiplication of a polynomial by a scalar, forms a linear space. Show that the set of all polynomials with real coefficients of degree less than or equal to $n$, under the ordinary operations of addition of polynomials and multiplication of a polynomial by a scalar, forms a linear space.

**Exercise**. (** Function Space**) Show that the set $F(x)$ of all functions that map the real numbers into itself is a linear space. Show that the set $F[a,b]$ of all functions on the interval $[a,b]$ using the standard operations is a linear space.

**Exercise**. (** The Space of Infinite Sequences**) Show that the set of all infinite sequences of real numbers is a linear space, where addition and scale multiplication are defined term by term.

**Exercise**. (** The Space of Linear Equations**) Show that the set $L_n$ of all linear equations with $n$ variables, forms a linear space.

**Exercise**. Let $\mathbb{V}$ be a linear space with $u\in V$ and let $a$ and $b$ be scalars. Prove that if $a u=bu$ and $u\neq 0$, then $a=b.$