# 1.2 Vector Spaces

In Section 1.1, we saw that with the natural definitions of vector addition and scalar multiplication, the vectors in a plane satisfy the eight properties listed on page 3. Many other familiar algebraic systems also permit definitions of addition and scalar multiplication that satisfy the same eight properties. In this section, we introduce some of these systems, but first we formally define this type of algebraic structure.

# Definitions.

*A vector space (or linear space)* V

*over a field*

^{2}

`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*

`ax`

*in*V,

*such that the following conditions hold*.

(VS 1)

*For all*V, $x+y=y+x$`x`,`y`in*(commutativity of addition).*(VS 2)

*For all*V, $(x+y)+z=x+(y+z)$`x`,`y`,`z`in*(associativity of addition).*(VS 3)

*There exists an element in*V*denoted by 0 such that*$x+0=x$*for each*V.`x`in(VS 4)

*For each element*V`x`in*there exists an element*V`y`in*such that*$x+y=0$.(VS 5)

*For each element*V, $1x=x$.`x`in(VS 6)

*For each pair of elements*V, $(ab)x=a(bx)$.`a`,`b`in`F`and each element`x`in(VS 7)

*For each element a in*V, $a(x+y)=ax+ay$.`F`and each pair of elements`x`,`y`in(VS 8)

*For each pair of elements*V, $(a+b)x=ax+bx$.`a`,`b`in`F`and each element`x`in

*The elements* $x+y$ *and ax 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**. *The reader should not confuse this use of the word “vector” with the physical entity discussed in Section 1.1: the word “vector” is now being used to describe any element of a vector space.*

A vector space is frequently discussed in the text without explicitly mentioning its field of scalars. The reader is cautioned to remember, however, that *every vector space is regarded as a vector space over a given Geld, which is denoted by* F. Occasionally we restrict our attention to the fields of real and complex numbers, which are denoted *R* and C, respectively. Unless otherwise noted, we assume that fields used in the examples and exercises of this book have characteristic zero (see page 549).

Observe that (VS 2) permits us to define the addition of any finite number of vectors unambiguously (without the use of parentheses).

In the remainder of this section we introduce several important examples of vector spaces that are studied throughout this text. Observe that in describing a vector space, it is necessary to specify not only the vectors but also the operations of addition and scalar multiplication. The reader should check that each of these examples satisfies conditions (VS1) through (VS8).

An object of the form $({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n})$, where the entries ${a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n}$ are elements of a field `F`, is called an ** n-tuple** with entries from

`F`. The elements ${a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n}$ are called the entries or components of the

`n`-tuple. Two

`n`-tuples $({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n})$ and $({b}_{1},\text{}{b}_{2},\text{}\dots ,\text{}{b}_{n})$ with entries from a field

`F`are called

**equal**if ${a}_{i}={b}_{i}$ for $i=1,\text{}2,\text{}\dots ,\text{}n$.

# Example 1

The set of all `n`-tuples with entries from a field `F` is denoted by ${\text{F}}^{n}$. This set is a vector space over `F` with the operations of coordinatewise addition and scalar multiplication; that is, if $u=({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n})\in {\text{F}}^{n},\text{}v=({b}_{1},\text{}{b}_{2},\text{}\dots ,\text{}{b}_{n})\in {\text{F}}^{n}$, and $c\in F$, then

Thus ${\text{R}}^{3}$ is a vector space over `R`. In this vector space,

Similarly, ${\text{C}}^{2}$ is a vector space over `C`. In this vector space,

Vectors in ${\text{F}}^{n}$ may be written as column vectors

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

An $m\times n$ **matrix** with entries from a field `F` is a rectangular array of the form

where each entry ${a}_{ij}\text{}(1\le i\le m,\text{}1\le j\le n)$ is an element of `F`. We call the entries ${a}_{ij}$ with $i=j$ the diagonal entries of the matrix. The entries ${a}_{i1},\text{}{a}_{i2},\text{}\dots ,\text{}{a}_{in}$ compose the ** ith row** of the matrix, and the entries ${a}_{1j},\text{}{a}_{2j},\text{}\dots ,\text{}{a}_{mj}$ compose the

*j*th column of the matrix. The rows of the preceding matrix are regarded as vectors in ${\text{F}}^{n}$, and the columns are regarded as vectors in ${\text{F}}^{m}$. Furthermore, we may regard a row vector in ${\text{F}}^{n}$ as a $1\times n$ matrix with entries from

`F`, and we may regard a column vector in ${\text{F}}^{m}$ as an $m\times 1$ matrix with entries from

`F`.

The $m\times n$ matrix in which each entry equals zero is called the **zero matrix** and is denoted by `O`.

In this book, we denote matrices by capital italic letters (e.g., `A`, `B`, and `C`), and we denote the entry of a matrix `A` that lies in row `i` and column `j` by ${A}_{ij}$. 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 ${A}_{ij}={B}_{ij}$ for $1\le i\le m$ and $1\le j\le n$.

# Example 2

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

for $1\le i\le m$ and $1\le j\le n$. For instance,

and

in ${\text{M}}_{2\times 3}(R)$.

Notice that the definitions of matrix addition and scalar multiplication in ${\text{M}}_{m\times n}(F)$ are natural extensions of the corresponding operations in ${\text{F}}^{n}$ and ${\text{F}}^{m}$. Thus the sum of two $m\times n$ matrices `A` and `B` in ${\text{M}}_{m\times n}(F)$ is the matrix in ${\text{M}}_{m\times n}(F)$ whose `i`th row vector is the sum of the `i`th row vectors of `A` and `B`, and, for any scalar `c`, the matrix `cA` is the matrix in ${\text{M}}_{m\times n}(F)$ whose `i`th row vector is `c` times the `i`th row vector of `A`. Likewise, the sum of two matrices `A` and `B` in ${\text{M}}_{m\times n}(F)$ is the matrix in ${\text{M}}_{m\times n}(F)$ whose `j`th column is the sum of the `j`th column vectors of `A` and `B`, and, for any scalar `c`, the matrix `cA` is the matrix in ${\text{M}}_{m\times n}(F)$ whose `j`th column vector is `c` times the `j`th column vector of `A`.

# Example 3

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

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

A polynomial with coefficients from a field `F` is an expression of the form

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

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,

and

are called **equal** if $m=n$ and ${a}_{i}={b}_{i}$ for $i=0,\text{}1,\text{}\dots ,\text{}n$.

When `F` is a field containing infinitely many scalars, we usually regard a polynomial with coefficients from `F` as a function from `F` into `F`. (See page 564.) In this case, the value of the function

at $c\in F$ is the scalar

Here either of the notations `f` or `f`(`x`) is used for the polynomial function

# Example 4

Let

and

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

Define

and for any $c\in F$, define

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 P(`F`).

We will see later that P(`F`) is essentially the same as a subset of the vector space defined in the next example.

# Example 5

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,\text{}2,\text{}\dots $ is denoted $({a}_{n})$. Let V consist of all the sequences $({a}_{n})$ in `F`. For $({a}_{n})$ and $({b}_{n})$ in V and $t\in F$, define

With these operations V is a vector space.

Our next two examples contain sets on which addition and scalar multiplication are defined, but which are *not* vector spaces.

# Example 6

Let $S=\{({a}_{1},\text{}{a}_{2}):\text{}{a}_{1},\text{}{a}_{2}\in R\}$. For $({a}_{1},\text{}{a}_{2}),\text{}({b}_{1},\text{}{b}_{2})\in S$ and $c\in R$, define

Since (VS 1), (VS 2), and (VS 8) fail to hold, S is not a vector space with these operations.

# Example 7

Let `S` be as in Example 6. For $({a}_{1},\text{}{a}_{2}),\text{}({b}_{1},\text{}{b}_{2})\in S$ and $c\in R$, define

Then `S` is not a vector space with these operations because (VS 3) (hence (VS 4)) and (VS 5) fail.

We conclude this section with a few of the elementary consequences of the definition of a vector space.

# Theorem 1.1 (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 $z+u=0$ (VS 4). Thus

by (VS 2) and (VS 3).

# Corollary 1.

*The vector 0 described in (VS 3) is unique.*

Proof. Exercise.

# Corollary 2.

*The vector* `y` *described in (VS 4) is unique.*

Proof. Exercise.

The vector *0* in (VS 3) is called the zero vector of V, and the vector `y` in (VS 4) (that is, the unique vector such that $x+y=0$) is called the **additive inverse** of `x` and is denoted by $-x$.

The next result contains some of the elementary properties of scalar multiplication.

# Theorem 1.2.

*In any vector space* V, *the following statements are true:*

(a) $0x=0$

*for each*$x\in \text{V}$.(b) $\left(-a\right)x=-\left(ax\right)=a\left(-x\right)$

*for each*$a\in F$*and each*$x\in \text{V}$.(c) $a0=0$

*for each*$a\in F$.

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

Hence $0x=0$ by Theorem 1.1.

(b) The vector $-\left(ax\right)$ is the unique element of V such that $ax+\left[-\left(ax\right)\right]=0$. Thus if $ax+\left(-a\right)x=0$, Corollary 2 to Theorem 1.1 implies that $\left(-a\right)x=-\left(ax\right)$. But by (VS 8),

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

The proof of (c) is similar to the proof of (a).

# Exercises

Label the following statements as true or false.

(a) Every vector space contains a zero vector.

(b) A vector space may have more than one zero vector.

(c) In any vector space, $ax=bx$ implies that $a=b$.

(d) In any vector space, $ax=ay$ implies that $x=y$.

(e) A vector in ${\text{F}}^{n}$ may be regarded as a matrix in ${\text{M}}_{n\times 1}\left(F\right)$.

(f) An $m\times n$ matrix has

`m`columns and`n`rows.(g) In P(

`F`), only polynomials of the same degree may be added.(h) If

`f`and`g`are polynomials of degree`n`, then $f+g$ is a polynomial of degree`n`.(i) If

`f`is a polynomial of degree`n`and`c`is a nonzero scalar, then`cf`is a polynomial of degree`n`.(j) A nonzero scalar of

`F`may be considered to be a polynomial in P(`F`) having degree zero.(k) Two functions in $\mathcal{F}\left(S,\text{}F\right)$ are equal if and only if they have the same value at each element of

`S`.

Write the zero vector of ${\text{M}}_{3\times 4}\left(F\right)$.

If

$$M=\left(\begin{array}{rrr}1& 2& 3\\ 4& 5& 6\end{array}\right),$$what are ${M}_{13},\text{}{M}_{21}$ and ${M}_{22}$?

Perform the indicated operations.

(a) $\left(\begin{array}{rrr}2& 5& -3\\ 1& 0& 7\end{array}\right)+\left(\begin{array}{rrr}4& -2& 5\\ -5& 3& 2\end{array}\right)$

(b) $\left(\begin{array}{rr}-6& 4\\ 3& -2\\ 1& 8\end{array}\right)+\left(\begin{array}{rr}7& -5\\ 0& -3\\ 2& 0\end{array}\right)$

(c) $4\left(\begin{array}{rrr}2& 5& -3\\ 1& 0& 7\end{array}\right)$

(d) $-5\left(\begin{array}{rr}-6& 4\\ 3& -2\\ 1& 8\end{array}\right)$

(e) $\left(2{x}^{4}-7{x}^{3}+4x+3\right)+\left(8{x}^{3}+2{x}^{2}-6x+7\right)$

(f) $\left(-3{x}^{3}+7{x}^{2}+8x-6\right)+\left(2{x}^{3}-8x+10\right)$

(g) $5\left(2{x}^{7}-6{x}^{4}+8{x}^{2}-3x\right)$

(h) $3\left({x}^{5}-2{x}^{3}+4x+2\right)$

Exercises 5 and 6 show why the definitions of matrix addition and scalar multiplication (as defined in Example 2) are the appropriate ones.

Richard Gard (“E.ects of Beaver on Trout in Sagehen Creek, California,”

*J. Wildlife Management*,**25**, 221-242) reports the following number of trout having crossed beaver dams in Sagehen Creek.# Upstream Crossings

Fall Spring Summer Brook trout 8 3 1 Rainbow trout 3 0 0 Brown trout 3 0 0 # Downstream Crossings

Fall Spring Summer Brook trout 9 1 4 Rainbow trout 3 0 0 Brown trout 1 1 0 Record the upstream and downstream crossings in two $3\times 3$ matrices, and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season.

At the end of May, a furniture store had the following inventory.

Early American Spanish Mediterranean Danish Living room suites 4 2 1 3 Bedroom suites 5 1 1 4 Dining room suites 3 1 2 6 Record these data as a $3\times 4$ matrix

*M*. To prepare for its June sale, the store decided to double its inventory on each of the items listed in the preceding table. Assuming that none of the present stock is sold until the additional furniture arrives, verify that the inventory on hand after the order is filled is described by the matrix 2`M`. If the inventory at the end of June is described by the matrix$$A=\left(\begin{array}{rrrr}5& 3& 1& 2\\ 6& 2& 1& 5\\ 1& 0& 3& 3\end{array}\right),$$interpret $2M-A$. How many suites were sold during the June sale?

Let $S=\left\{0,\text{}1\right\}$ and $F=R$. In $\mathcal{F}(S,\text{}R)$, show that $f=g$ and $f+g=h$, where $f(t)=2t+1,\text{}g(t)=1+4t-2{t}^{2}$, and $h\left(t\right)={5}^{t}+1$.

In any vector space V, show that $\left(a+b\right)\left(x+y\right)=ax+ay+bx+by$ for any $x,\text{}y\in \text{V}$ and any $a,\text{}b\in F$.

Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). Visit goo.gl/

WFWgzX for a solution.Let V denote the set of all differentiable real-valued functions defined on the real line. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

Let $\text{V}=\left\{0\right\}$ consist of a single vector

*0*and define $0+0=0$ and $c0=0$ for each scalar`c`in`F`. Prove that V is a vector space over`F`. (V is called the**zero vector space**.)A real-valued function

`f`defined on the real line is called an even function if $f\left(-t\right)=f\left(t\right)$ for each real number`t`. Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space.Let V denote the set of ordered pairs of real numbers. If $\left({a}_{1},\text{}{a}_{2}\right)$ and $\left({b}_{1},\text{}{b}_{2}\right)$ are elements of V and $c\in R$, define

$$\left({a}_{1},\text{}{a}_{2}\right)+\left({b}_{1},\text{}{b}_{2}\right)=\left({a}_{1}+{b}_{1},\text{}{a}_{2}{b}_{2}\right)\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}c({a}_{1},\text{}{a}_{2})=(c{a}_{1,}c{a}_{2}).$$Is V a vector space over

*R*with these operations? Justify your answer.Let $\text{V}=\{({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n}):\text{}{a}_{i}\in C\text{for}i=1,\text{}2,\text{}\dots ,\text{}n\};$ so V is a vector space over

`C`by Example 1. Is V a vector space over the field of real numbers with the operations of coordinatewise addition and multiplication?Let $\text{V}=\left\{\left({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{n}\right):\text{}{a}_{i}\in R\text{for}i=1,\text{}2,\text{}\dots ,\text{}n\right\};$ so V is a vector space over

`R`by Example 1. Is V a vector space over the field of complex numbers with the operations of coordinatewise addition and multiplication?Let V denote the set of all $m\times n$ matrices with real entries; so V is a vector space over

`R`by Example 2. Let`F`be the field of rational numbers. Is V a vector space over`F`with the usual definitions of matrix addition and scalar multiplication?Let $\text{V}=\left\{\left({a}_{1},\text{}{a}_{2}\right):{a}_{1},\text{}{a}_{2}\in F\right\}$, where

`F`is a field. Define addition of elements of V coordinatewise, and for $c\in F$ and $\left({a}_{1},\text{}{a}_{2}\right)\in \text{V}$, define$$c\left({a}_{1},\text{}{a}_{2}\right)=\left({a}_{1},\text{}0\right).$$Is V a vector space over

`F`with these operations? Justify your answer.Let $\text{V}=\left\{\left({a}_{1},\text{}{a}_{2}\right):\text{}{a}_{1},\text{}{a}_{2}\in R\right\}$. For $\left({a}_{1},\text{}{a}_{2}\right),\text{}\left({b}_{1},\text{}{b}_{2}\right)\in V$ and $c\in R$, define

$$\left({a}_{1},\text{}{a}_{2}\right)+\left({b}_{1},\text{}{b}_{2}\right)=\left({a}_{1}+2{b}_{1},\text{}{a}_{2}+3{b}_{2}\right)\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}c\left({a}_{1},\text{}{a}_{2}\right)=\left(c{a}_{1},\text{}c{a}_{2}\right).$$Is V a vector space over

*R*with these operations? Justify your answer.Let $\text{V}=\left\{\left({a}_{1},\text{}{a}_{2}\right):\text{}{a}_{1},\text{}{a}_{2}\in R\right\}$. Define addition of elements of V coordinatewise, and for $\left({a}_{1},\text{}{a}_{2}\right)$ in V and $c\in R$, define

$$c({a}_{1},\text{}{a}_{2})=\{\begin{array}{ll}(0,\text{}0)& \text{if}c=0\\ \left(c{a}_{1},\text{}{\displaystyle {\displaystyle \frac{{a}_{2}}{c}}}\right)& \text{if}c\ne 0.\end{array}$$Is V a vector space over

*R*with these operations? Justify your answer.Let V denote the set of all real-valued functions

`f`defined on the real line such that $f\left(1\right)=0$. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.Let V and W be vector spaces over a field

`F`. Let$$\text{Z=}\left\{(v,\text{}w):\text{}v\in \text{Vand}w\in \text{W}\right\}.$$Prove that

`Z`is a vector space over`F`with the operations$$({v}_{1},\text{}{w}_{1})+({v}_{2},\text{}{w}_{2})=({v}_{1}+{v}_{2},\text{}{w}_{1}+{w}_{2})\phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1.5em}{0ex}}c({v}_{1},\text{}{w}_{1})=(c{v}_{1},\text{}c{w}_{1}).$$-
How many matrices are there in the vector space ${\text{M}}_{m\times n}({Z}_{2})$? (See Appendix C.)