1.2 Vector Spaces – Linear Algebra, 5th Edition

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 field2 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 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, 1x=x.

  • (VS 6) For each pair of elements a, b in F and each element x in V, (ab)x=a(bx).

  • (VS 7) For each element a in F and each pair of elements x, y in V, a(x+y)=ax+ay.

  • (VS 8) For each pair of elements a, b in F and each element x in V, (a+b)x=ax+bx.

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 (a1, a2, , an), where the entries a1, a2, , an are elements of a field F, is called an n-tuple with entries from F. The elements a1, a2, , an are called the entries or components of the n-tuple. Two n-tuples (a1, a2, , an) and (b1, b2, , bn) with entries from a field F are called equal if ai=bi for i=1, 2, , n.

Example 1

The set of all n-tuples with entries from a field F is denoted by Fn. This set is a vector space over F with the operations of coordinatewise addition and scalar multiplication; that is, if u=(a1, a2, , an)Fn, v=(b1, b2, , bn)Fn, and cF, then

u+v=(a1+b1, a2+b2, , an+bn)andcu=(ca1, ca2, , can).

Thus R3 is a vector space over R. In this vector space,

(3,2, 0)+(1, 1, 4)=(2,1, 4)and5(1,2, 0)=(5, 10, 0).

Similarly, C2 is a vector space over C. In this vector space,

(1+i, 2)+(23i, 4i)=(32i, 2+4i)andi(1+i, 2)=(1+i, 2i).

Vectors in Fn may be written as column vectors

(a1a2an)

rather than as row vectors (a1, a2, , an). Since a 1-tuple whose only entry is from F can be regarded as an element of F, we usually write F rather than F1 for the vector space of 1-tuples with entry from F.

An m×n matrix with entries from a field F is a rectangular array of the form

(a11a12a1na21a22a2nam1am2amn),

where each entry aij (1im, 1jn) is an element of F. We call the entries aij with i=j the diagonal entries of the matrix. The entries ai1, ai2, , ain compose the ith row of the matrix, and the entries a1j, a2j, , amj compose the jth column of the matrix. The rows of the preceding matrix are regarded as vectors in Fn, and the columns are regarded as vectors in Fm. Furthermore, we may regard a row vector in Fn as a 1×n matrix with entries from F, and we may regard a column vector in Fm as an m×1 matrix with entries from F.

The m×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 Aij. In addition, if the number of rows and columns of a matrix are equal, the matrix is called square.

Two m×n matrices A and B are called equal if all their corresponding entries are equal, that is, if Aij=Bij for 1im and 1jn.

Example 2

The set of all m×n matrices with entries from a field F is a vector space, which we denote by Mm×n(F), with the following operations of matrix addition and scalar multiplication: For A, BMm×n(F) and cF,

(A+B)ij=Aij+Bijand(cA)ij=cAij

for 1im and 1jn. For instance,

(201134)+(526341)=(325413)

and

3(102323)=(306969)

in M2×3(R).

Notice that the definitions of matrix addition and scalar multiplication in Mm×n(F) are natural extensions of the corresponding operations in Fn and Fm. Thus the sum of two m×n matrices A and B in Mm×n(F) is the matrix in Mm×n(F) whose ith row vector is the sum of the ith row vectors of A and B, and, for any scalar c, the matrix cA is the matrix in Mm×n(F) whose ith row vector is c times the ith row vector of A. Likewise, the sum of two matrices A and B in Mm×n(F) is the matrix in Mm×n(F) whose jth column is the sum of the jth column vectors of A and B, and, for any scalar c, the matrix cA is the matrix in Mm×n(F) whose jth column vector is c times the jth column vector of A.

Example 3

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

(f+g)(s)=f(s)+g(s)and(cf)(s)=c[f(s)]

for each sS. 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

f(x)=anxn+an1xn1++a1x+a0,

where n is a nonnegative integer and each ak, called the coefficient of xk, is in F. If f(x)=0, that is, if an=an1==a0=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)=anxn+an1xn1++a1x+a0

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)=anxn+an1xn1++a1x+a0

and

g(x)=bmxm+bm1xm1++b1x+b0,

are called equal if m=n and ai=bi for i=0, 1, , 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

f(x)=anxn+an1xn1++a1x+a0

at cF is the scalar

f(c)=ancn+an1cn1++a1c+a0.

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

f(x)=anxn+an1xn1++a1x+a0.

Example 4

Let

f(x)=anxn+an1xn1++a1x+a0

and

g(x)=bmxm+bm1xm1++b1x+b0

be polynomials with coefficients from a field F. Suppose that mn, and define bm+1=bm+2==bn=0. Then g(x) can be written as

g(x)=bnxn+bn1xn1++b1x+b0.

Define

f(x)+g(x)=(an+bn)xn+(an1+bn1)xn1++(a1+b1)x+(a0+b0)

and for any cF, define

cf(x)=canxn+can1xn1++ca1x+ca0.

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 σ from the positive integers into F. In this book, the sequence σ such that σ(n)=an for n=1, 2,  is denoted (an). Let V consist of all the sequences (an) in F. For (an) and (bn) in V and tF, define

(an)+(bn)=(an+bn)andt(an)=(tan).

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={ (a1, a2): a1, a2R }. For (a1, a2), (b1, b2)S and cR, define

(a1, a2)+(b1, b2)=(a1+b1, a2b2)andc(a1, a2)=(ca1, ca2).

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 (a1, a2), (b1, b2)S and cR, define

(a1, a2)+(b1, b2)=(a1+b1, 0)andc(a1, a2)=(ca1, 0).

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

x=x+0=x+(z+v)=(x+z)+v=(y+z)+v=y+(z+v)=y+0=y

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:

  1. (a) 0x=0 for each xV.

  2. (b) (a)x=(ax)=a(x) for each aF and each xV.

  3. (c) a0=0 for each aF.

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

0x+0x=(0+0)x=0x=0x+0=0+0x.

Hence 0x=0 by Theorem 1.1.

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

ax+(a)x=[ a+(a) ]x=0x=0

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

a(x)=a[ (1)x ]=[ a(1) ]x=(a)x.

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

Exercises

  1. Label the following statements as true or false.

    1. (a) Every vector space contains a zero vector.

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

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

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

    5. (e) A vector in Fn may be regarded as a matrix in Mn×1(F).

    6. (f) An m×n matrix has m columns and n rows.

    7. (g) In P(F), only polynomials of the same degree may be added.

    8. (h) If f and g are polynomials of degree n, then f+g is a polynomial of degree n.

    9. (i) If f is a polynomial of degree n and c is a nonzero scalar, then cf is a polynomial of degree n.

    10. (j) A nonzero scalar of F may be considered to be a polynomial in P(F) having degree zero.

    11. (k) Two functions in F(S, F) are equal if and only if they have the same value at each element of S.

  2. Write the zero vector of M3×4(F).

  3. If

    M=(123456),

    what are M13, M21 and M22?

  4. Perform the indicated operations.

    1. (a) (253107)+(425532)

    2. (b) (643218)+(750320)

    3. (c) 4(253107)

    4. (d) 5(643218)

    5. (e) (2x47x3+4x+3)+(8x3+2x26x+7)

    6. (f) (3x3+7x2+8x6)+(2x38x+10)

    7. (g) 5(2x76x4+8x23x)

    8. (h) 3(x52x3+4x+2)

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

  1. 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 3×3 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.

  2. 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 3×4 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 2M. If the inventory at the end of June is described by the matrix

    A=(531262151033),

    interpret 2MA. How many suites were sold during the June sale?

  3. Let S={ 0, 1 } and F=R. In F(S, R), show that f=g and f+g=h, where f(t)=2t+1, g(t)=1+4t2t2, and h(t)=5t+1.

  4. In any vector space V, show that (a+b)(x+y)=ax+ay+bx+by for any x, yV and any a, bF.

  5. Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). Visit goo.gl/WFWgzX for a solution.

  6. 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.

  7. Let V={ 0 } 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.)

  8. A real-valued function f defined on the real line is called an even function if f(t)=f(t) 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.

  9. Let V denote the set of ordered pairs of real numbers. If (a1, a2) and (b1, b2) are elements of V and cR, define

    (a1, a2)+(b1, b2)=(a1+b1, a2b2)andc(a1, a2)=(ca1,ca2).

    Is V a vector space over R with these operations? Justify your answer.

  10. Let V={(a1, a2, , an): aiC for i=1, 2, , 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?

  11. Let V={ (a1, a2, , an): aiR for i=1, 2, , n }; 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?

  12. Let V denote the set of all m×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?

  13. Let V={ (a1, a2):a1, a2F }, where F is a field. Define addition of elements of V coordinatewise, and for cF and (a1, a2)V, define

    c(a1, a2)=(a1, 0).

    Is V a vector space over F with these operations? Justify your answer.

  14. Let V={ (a1, a2): a1, a2R }. For (a1, a2), (b1, b2)V and cR, define

    (a1, a2)+(b1, b2)=(a1+2b1, a2+3b2)andc(a1, a2)=(ca1, ca2).

    Is V a vector space over R with these operations? Justify your answer.

  15. Let V={ (a1, a2): a1, a2R }. Define addition of elements of V coordinatewise, and for (a1, a2) in V and cR, define

    c(a1, a2)={(0, 0)if c=0(ca1, a2c)if c0.

    Is V a vector space over R with these operations? Justify your answer.

  16. Let V denote the set of all real-valued functions f defined on the real line such that f(1)=0. Prove that V is a vector space with the operations of addition and scalar multiplication defined in Example 3.

  17. Let V and W be vector spaces over a field F. Let

    Z = { (v, w): vV and wW }.

    Prove that Z is a vector space over F with the operations

    (v1, w1)+(v2, w2)=(v1+v2, w1+w2)andc(v1, w1)=(cv1, cw1).
  18. How many matrices are there in the vector space Mm×n(Z2)? (See Appendix C.)