§ 2. Vector Spaces

Definition and Examples of Vector Space

Definition 2.1 : Vector Space

A vector space (or linear space) \(V\) over a field \(F\) consists of a set on which two operations (called addition and scalar multiplication, respectively) are defined so that for each pair of elements \(x, y\), in \(V\) there is a unique element \(x+y\) in \(V\), and for each element \(a\) in \(F\) and each element \(x\) in \(V\) there is a unique element \(a x\) in \(V\), such that the following conditions hold.

• (VS 1) For all \(x, y\) in \(V\), \(x+y=y+x\) (commutativity of addition).
• (VS 2) For all \(x, y, z\) in \(V\), \((x+y)+z=x+(y+z)\) (associativity of addition).
• (VS 3) There exists an element in \(V\) denoted by \(0\) such that \(x+0=x\) for each \(x\) in \(V\).
• (VS 4) For each element \(x\) in \(V\) there exists an element \(y\) in \(V\) such that \(x+y=0\).
• (VS 5) For each element \(x\) in \(V\), \(1 x=x\).
• (VS 6) For each pair of elements \(a, b\) in \(F\) and each element \(x\) in \(V\), \((a b) x=a(b x)\).
• (VS 7) For each element \(a\) in \(F\) and each pair of elements \(x, y\) in \(V\), \(a(x+y)=a x+a y\).
• (VS 8) For each pair of elements \(a, b\) in \(F\) and each element \(x\) in \(V\), \((a+b) x=a x+b x\).

The elements \(x+y\) and \(a x\) are called the sum of \(x\) and \(y\) and the product of \(a\) and \(x\), respectively.

The elements of the field \(F\) are called scalars and the elements of the vector space \(V\) are called vectors.

Example 2.2 : \(F^n\) (set of all \(n\)-tuples with entries from a field \(F\)) is a vector space.

An object of the form \((a_{1}, a_{2}, \ldots, a_{n})\), where the entries \(a_{1}, a_{2}, \ldots, a_{n}\) are elements of a field \(F\), is called an \(n\)-tuple with entries from \(F\). The elements \(a_{1}, a_{2}, \ldots a_{n}\) are called the entries or components of the \(n\)-tuple. Two \(n\)-tuples \((a_{1}, a_{2}, \ldots, a_{n})\) and \((b_{1}, b_{2}, \ldots, b_{n})\) with entries from a field \(F\) are called equal if \(a_{i}=b_{i}\) for \(i=1, 2, \ldots, n\).

---

The set of all \(n\)-tuples with entries from a field \(F\) is denoted by \(F^{n}\). This set is a vector space over \(F\) with the operations of coordinatewise addition and scalar multiplication: that is, if \(u=\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in F^n, v=\left(b_{1}, b_{2}, \ldots, b_{n}\right) \in F^n\), and \(c \in F\), then

\[ u+v=\left(a_{1}+b_{1}, a_{2}+b_{2}, \ldots, a_{n}+b_{n}\right) \quad \text { and } \quad c u=\left(c a_{1}, c a_{2}, \ldots, c a_{n}\right) . \]

Vectors in \(F^{n}\) may be written as column vectors

\[ \left(\begin{array}{c} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{array}\right) \]

rather than as row vectors \(\left(a_{1}, a_{2}, \ldots, a_{n}\right)\). Since a 1-tuple whose only entry is from \(F\) can be regarded as an element of \(F\), we usually write \(F\) rather than \(F^{1}\) for the vector space of 1-tuples with entry from \(F\).

Example 2.3 : \(\mathrm{M}_{m \times n}(F)\) (set of all \(m \times n\) matrices with entries from a field \(F\)) is a vector space.

An \(m \times n\) matrix with entries from a field \(F\) is a rectangular array of the form

\[ \left(\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \end{array}\right) \]

where each entry \(a_{i j}(1 \leq i \leq m, 1 \leq j \leq n)\) is an element of \(F\). We call the entries \(a_{i j}\) with \(i=j\) the diagonal entries of the matrix. The entries \(a_{i 1}, a_{i 2}, \ldots, a_{i n}\) compose the \(i\) th row of the matrix, and the entries \(a_{1 j}, a_{2 j}, \ldots , a_{m j}\) compose the \(j\) th column of the matrix. The rows of the preceding matrix are regarded as vectors in \(F^n\), and the columns are regarded as vectors in \(F^m\). The \(m \times n\) matrix in which each entry equals zero is called the zero matrix and is denoted by \(O\). In addition, if the number of rows and columns of a matrix are equal, the matrix is called square.

Two \(m \times n\) matrices \(A\) and \(B\) are called equal if all their corresponding entries are equal, that is, if \(A_{i j}=B_{i j}\) for \(1 \leq i \leq m\) and \(1 \leq j \leq n\).

---

The set of all \(m \times n\) matrices with entries from a field \(F\) is a vector space, which we denote by \(\mathrm{M}_{m \times n}(F)\), with the following operations of matrix addition and scalar multiplication: For \(A, B \in \mathrm{M}_{m \times n}(F)\) and \(c \in F\),

\[ (A+B)_{i j}=A_{i j}+B_{i j} \quad \text { and } \quad(c A)_{i j}=c A_{i j} \]

for \(1 \leq i \leq m\) and \(1 \leq j \leq n\).

Example 2.4 : \(\mathcal{F}(S, F)\) (set of all functions from \(S\) to \(F\)) is a vector space.

Let \(S\) be any nonempty set and \(F\) be any field, and let \(\mathcal{F}(S, F)\) denote the set of all functions from \(S\) to \(F\). Two functions \(f\) and \(g\) in \(\mathcal{F}(S, F)\) are called equal if \(f(s)=g(s)\) for each \(s \in S\). The set \(\mathcal{F}(S, F)\) is a vector space with the operations of addition and scalar multiplication defined for \(f, g \in \mathcal{F}(S, F)\) and \(c \in F\) by

\[ (f+g)(s)=f(s)+g(s) \quad \text { and } \quad(c f)(s)=c[f(s)] \]

for each \(s \in S\). Note that these are the familiar operations of addition and scalar multiplication for functions used in algebra and calculus.

Example 2.5 : \(\mathrm{P}(F)\) (set of all polynomials with coefficients from \(F\)) is a vector space.

A polynomial with coefficients from a field \(F\) is an expression of the form

\[ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

where \(n\) is a nonnegative integer and each \(a_{k}\), called the coefficient of \(x^{k}\), is in \(F\). If \(f(x)=0\), that is, if \(a_{n}=a_{n-1}=\cdots=a_{0}=0\), then \(f(x)\) is called the zero polynomial and, for convenience, its degree is defined to be \(-1\); otherwise, the degree of a polynomial is defined to be the largest exponent of \(x\) that appears in the representation

\[ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

with a nonzero coefficient. Note that the polynomials of degree zero may be written in the form \(f(x)=c\) for some nonzero scalar \(c\).

Two polynomials,

\[ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

and

\[ g(x)=b_{m} x^{m}+b_{m-1} x^{m-1}+\cdots+b_{1} x+b_{0} \]

are called equal if \(m=n\) and \(a_{i}=b_{i}\) for \(i=0,1, \ldots, n\).

---

Let

\[ f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \]

and

\[ g(x)=b_{m} x^{m}+b_{m-1} x^{m-1}+\cdots+b_{1} x+b_{0} \]

be polynomials with coefficients from a field \(F\). Suppose that \(m \leq n\), and define \(b_{m+1}=b_{m+2}=\cdots=b_{n}=0\). Then \(g(x)\) can be written as

\[ g(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0} \]

Define

\[ f(x)+g(x)=\left(a_{n}+b_{n}\right) x^{n}+\left(a_{n-1}+b_{n-1}\right) x^{n-1}+\cdots+\left(a_{1}+b_{1}\right) x+\left(a_{0}+b_{0}\right) \]

and for any \(c \in F\), define

\[ c f(x)=c a_{n} x^{n}+c a_{n-1} x^{n-1}+\cdots+c a_{1} x+c a_{0} \]

With these operations of addition and scalar multiplication, the set of all polynomials with coefficients from \(F\) is a vector space, which we denote by \(\mathrm{P}(F)\).

Example 2.6 : Sequence in \(F\) is a vector space.

Let \(F\) be any field. A sequence in \(F\) is a function \(\sigma\) from the positive integers into \(F\). In this book, the sequence \(\sigma\) such that \(\sigma(n)=a_{n}\) for \(n=1,2, \ldots\) is denoted \(\left\{a_{n}\right\}\).

Let \(V\) consist of all sequences \(\left\{a_{n}\right\}\) in \(F\) that have only a finite number of nonzero terms \(a_{n}\). If \(\left\{a_{n}\right\}\) and \(\left\{b_{n}\right\}\) are in \(V\) and \(t \in F\), define

\[ \left\{a_{n}\right\}+\left\{b_{n}\right\}=\left\{a_{n}+b_{n}\right\} \quad \text { and } \quad t\left\{a_{n}\right\}=\left\{t a_{n}\right\} . \]

With these operations \(V\) is a vector space.

Elementary Properties of Vector Space

Theorem 2.7 : Cancellation law for vector addition

If \(x, y\), and \(z\) are vectors in a vector space \(V\) such that \(x+z=y+z\), then \(x=y\).

Proof

There exists a vector \(v\) in \(V\) such that \(z+v=0\) (VS 4). Thus

\[ \begin{aligned} x & =x+0=x+(z+v)=(x+z)+v \\ & =(y+z)+v=y+(z+y)=y+0=y \end{aligned} \]

by (VS 2) and (VS 3).

Corollary 2.8 : Uniqueness of zero vector

The vector \(0\) described in (VS 3) (that is, the vector such that \(x+0=x\) for all \(x\)) is unique.

The unique vector \(0\) in (VS 3) is called the zero vector of \(V\).

Proof

Let \(x+0=x\) and \(x+0'=x\) for all \(x\). Then

\[ x=x+0=x+0' \]

and by Theorem 2.7, \(0=0'\).

Corollary 2.9 : Uniqueness of additive inverse

The vector \(y\) described in (VS 4) (that is, the vector such that \(x+y=0\) for a given \(x\)) is unique.

The unique vector \(y\) in (VS 4) is called the additive inverse of \(x\) and is denoted by \(-x\).

Proof

For a fixed \(x\), let \(x+y=0\) and \(x+y'=0\). Then

\[ 0=x+y=x+y' \]

and by Theorem 2.7, \(y=y'\).

Theorem 2.10 : Basic properties of scalar multiplication

In any vector space \(V\), the following statements are true:

• (a) \(0 x=0\) for each \(x \in V\).
• (b) \((-a) x=-(a x)=a(-x)\) for each \(a \in F\) and each \(x \in V\).
• (c) \(a 0=0\) for each \(a \in F\).

Proof

• (a)
  By (VS 8), (VS 3), and (VS 1), it follows that

  \[ 0 x+0 x=(0+0) x=0 x=0 x+0=0+0 x \]

  Hence \(0 x=0\) by Theorem 2.7.

• (b)
  The vector \(-(a x)\) is the unique element of \(V\) such that \(a x+[-(a x)]=0\). Thus if \(a x+(-a) x=0\). Corollary 2.9 implies that \((-a) x=-(a x)\). But by (VS 8),

  \[ a x+(-a) x=[a+(-a)] x=0 x=0 \]

  by (a). Consequently \((-a) x=-(a x)\). In particular, \((-1) x=-x\). So, by (VS 6),

  \[ a(-x)=a[(-1) x]=[a(-1)] x=(-a) x \]

• (c)
  By (VS 7), (VS 3), and (VS 1), it follows that

  \[ a0 + a0 = a(0+0) = a0 = a0 + 0 = 0 + a0 \]

  Hence \(a0 = 0\) by Theorem 2.7.