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.
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
(VS 1) For all x, y in V, x+y=y+x
(VS 2) For all x, y, z in V, (x+y)+z=x+(y+z)
(VS 3) There exists an element in V denoted by 0 such that x+0=x
(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
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)
The set of all n-tuples with entries from a field F is denoted by Fn
Thus R3
Similarly, C2
Vectors in Fn
rather than as row vectors (a1, a2, …, an)
An m×n
where each entry aij (1≤i≤m, 1≤j≤n)
The m×n
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
Two m×n
The set of all m×n
for 1≤i≤m
and
in M2×3(R)
Notice that the definitions of matrix addition and scalar multiplication in Mm×n(F)
Let S be any nonempty set and F be any field, and let F(S, F)
for each s∈S
A polynomial with coefficients from a field F is an expression of the form
where n is a nonnegative integer and each ak
with a nonzero coefficient. Note that the polynomials of degree zero may be written in the form f(x)=c
and
are called equal if m=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∈F
Here either of the notations f or f(x) is used for the polynomial function
Let
and
be polynomials with coefficients from a field F. Suppose that m≤n
Define
and for any c∈F
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.
Let F be any field. A sequence in F is a function σ
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.
Let S={(a1, a2): a1, a2∈R}
Since (VS 1), (VS 2), and (VS 8) fail to hold, S is not a vector space with these operations.
Let S be as in Example 6. For (a1, a2), (b1, b2)∈S
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.
If x, y, and z are vectors in a vector space V such that x+z=y+z
Proof. There exists a vector v in V such that z+u=0
by (VS 2) and (VS 3).
The vector 0 described in (VS 3) is unique.
Proof. Exercise.
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
The next result contains some of the elementary properties of scalar multiplication.
In any vector space V, the following statements are true:
(a) 0x=0
(b) (−a)x=−(ax)=a(−x)
(c) a0=0
Proof. (a) By (VS 8), (VS 3), and (VS 1), it follows that
Hence 0x=0
(b) The vector −(ax)
by (a). Consequently (−a)x=−(ax)
The proof of (c) is similar to the proof of (a).
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
(d) In any vector space, ax=ay
(e) A vector in Fn
(f) An m×n
(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
(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 F(S, F)
Write the zero vector of M3×4(F)
If
what are M13, M21
Perform the indicated operations.
(a) (25−3107)+(4−25−532)
(b) (−643−218)+(7−50−320)
(c) 4(25−3107)
(d) −5(−643−218)
(e) (2x4−7x3+4x+3)+(8x3+2x2−6x+7)
(f) (−3x3+7x2+8x−6)+(2x3−8x+10)
(g) 5(2x7−6x4+8x2−3x)
(h) 3(x5−2x3+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.
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.
Fall | Spring | Summer | |
---|---|---|---|
Brook trout | 8 | 3 | 1 |
Rainbow trout | 3 | 0 | 0 |
Brown trout | 3 | 0 | 0 |
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
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
interpret 2M−A
Let S={0, 1}
In any vector space V, show that (a+b)(x+y)=ax+ay+bx+by
Prove Corollaries 1 and 2 of Theorem 1.1 and Theorem 1.2(c). Visit goo.gl/
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 V={0}
A real-valued function f defined on the real line is called an even function if f(−t)=f(t)
Let V denote the set of ordered pairs of real numbers. If (a1, a2)
Is V a vector space over R with these operations? Justify your answer.
Let V={(a1, a2, …, an): ai∈C for i=1, 2, …, n};
Let V={(a1, a2, …, an): ai∈R for i=1, 2, …, n};
Let V denote the set of all m×n
Let V={(a1, a2):a1, a2∈F}
Is V a vector space over F with these operations? Justify your answer.
Let V={(a1, a2): a1, a2∈R}
Is V a vector space over R with these operations? Justify your answer.
Let V={(a1, a2): a1, a2∈R}
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(1)=0
Let V and W be vector spaces over a field F. Let
Prove that Z is a vector space over F with the operations
How many matrices are there in the vector space Mm×n(Z2)