INTRODUCTION

In the preceding sections we were dealing with points and vectors in 2- and 3-space. Mathematicians in the nineteenth century, notably the English mathematicians Arthur Cayley (1821–1895) and James Joseph Sylvester (1814–1897) and the Irish mathematician William Rowan Hamilton (1805–1865), realized that the concepts of point and vector could be generalized. A realization developed that vectors could be described, or defined, by analytic rather than geometric properties. This was a truly significant breakthrough in the history of mathematics. There is no need to stop with three dimensions; ordered quadruples 〈a₁, a₂, a₃, a₄〉, quintuples 〈a₁, a₂, a₃, a₄, a₅〉, and n-tuples 〈a₁, a₂, … , a_n〉 of real numbers can be thought of as vectors just as well as ordered pairs 〈a₁, a₂〉 and ordered triples 〈a₁, a₂, a₃〉, the only difference being that we lose our ability to visualize directed line segments or arrows in 4-dimensional, 5-dimensional, or n-dimensional space.

n-Space

In formal terms, a vector in n-space is any ordered n-tuple a = 〈a₁, a₂, … , a_n〉 of real numbers called the components of a. The set of all vectors in n-space is denoted by Rⁿ. The concepts of vector addition, scalar multiplication, equality, and so on listed in Definition 7.2.1 carry over to Rⁿ in a natural way. For example, if a = 〈a₁, a₂, … , a_n〉 and b = 〈b₁, b₂, … , b_n〉, then addition and scalar multiplication in n-space are defined by

a + b = 〈a₁ + b₁, a₂ + b₂, … , a_n + b_n〉 and ka = 〈ka₁, ka₂, … , ka_n〉. (1)

The zero vector in Rⁿ is 0 = 〈0, 0, … , 0〉. The notion of length or magnitude of a vector a = 〈a₁, a₂, … , a_n〉 in n-space is just an extension of that concept in 2- and 3-space:

.

The length of a vector is also called its norm. A unit vector is one whose norm is 1. For a nonzero vector a, the process of constructing a unit vector u by multiplying a by the reciprocal of its norm, that is, u = , is referred to as normalizing a. For example, if a = 〈3, 1, 2, −1〉, then and a unit vector is

The standard inner product, also known as the Euclidean inner product or dot product, of two n-vectors a = 〈a₁, a₂, … , a_n〉 and b = 〈b₁, b₂, … , b_n〉 is the real number defined by

(2)

Two nonzero vectors a and b in Rⁿ are said to be orthogonal if and only if a · b = 0. For example, a = 〈3, 4, 1, −6〉 and b = 〈1, , 1, 1〉 are orthogonal in R⁴ since a · b = 3 · 1 + 4 · + 1 · 1 + (−6) · 1 = 0.

Vector Space

We can even go beyond the notion of a vector as an ordered n-tuple in Rⁿ. A vector can be defined as anything we want it to be: an ordered n-tuple, a number, an array of numbers, or even a function. But we are particularly interested in vectors that are elements in a special kind of set called a vector space. Fundamental to the notion of vector space are two kinds of objects, vectors and scalars, and two algebraic operations analogous to those given in (1). For a set of vectors we want to be able to add two vectors in this set and get another vector in the same set, and we want to multiply a vector by a scalar and obtain a vector in the same set. To determine whether a set of objects is a vector space depends on whether the set possesses these two algebraic operations along with certain other properties. These properties, the axioms of a vector space, are given next.

In this brief introduction to abstract vectors we shall take the scalars in Definition 7.6.1 to be real numbers. In this case V is referred to as a real vector space, although we shall not belabor this term. When the scalars are allowed to be complex numbers we obtain a complex vector space. Since properties (i)–(viii) on page 331 are the prototypes for the axioms in Definition 7.6.1, it is clear that R² is a vector space. Moreover, since vectors in R³ and Rⁿ have these same properties, we conclude that R³ and Rⁿ are also vector spaces. Axioms (i) and (vi) are called the closure axioms and we say that a vector space V is closed under vector addition and scalar multiplication. Note, too, that concepts such as length and inner product are not part of the axiomatic structure of a vector space.

EXAMPLE 1 Checking the Closure Axioms

The vector space V = {0} is often called the trivial or zero vector space.

If this is your first experience with the notion of an abstract vector, then you are cautioned to not take the names vector addition and scalar multiplication too literally. These operations are defined and as such you must accept them at face value even though these operations may not bear any resemblance to the usual understanding of ordinary addition and multiplication in, say, R, R², R³, or Rⁿ. For example, the addition of two vectors x and y could be x − y. With this forewarning, consider the next example.

EXAMPLE 2 An Example of a Vector Space

Here are some important vector spaces—we have mentioned some of these previously. The operations of vector addition and scalar multiplication are the usual operations associated with the set.

• The set R of real numbers
• The set R² of ordered pairs
• The set R³ of ordered triples
• The set Rⁿ of ordered n-tuples
• The set P_n of polynomials of degree less than or equal to n
• The set P of all polynomials
• The set of real-valued functions f defined on the entire real line
• The set C[a, b] of real-valued functions f continuous on the closed interval [a, b]
• The set C(−, ) of real-valued functions f continuous on the entire real line
• The set Cⁿ[a, b] of all real-valued functions f for which f, f′ ,f″ , …, f⁽ⁿ⁾ exist and are continuous on the closed interval [a, b]

Subspace

It may happen that a subset of vectors W of a vector space V is itself a vector space.

Every vector space V has at least two subspaces: V itself and the zero subspace {0}; {0} is a subspace since the zero vector must be an element in every vector space.

To show that a subset W of a vector space V is a subspace, it is not necessary to demonstrate that all 10 axioms of Definition 7.6.1 are satisfied. Since all the vectors in W are also in V, these vectors must satisfy axioms such as (ii) and (iii). In other words, W inherits most of the properties of a vector space from V. As the next theorem indicates, we need only check the two closure axioms to demonstrate that a subset W is a subspace of V.

EXAMPLE 3 A Subspace

EXAMPLE 4 A Subspace

It is always a good idea to have concrete visualizations of vector spaces and subspaces. The subspaces of the vector space R³ of three-dimensional vectors can be easily visualized by thinking of a vector as a point (a₁, a₂, a₃). Of course, {0} and R³ itself are subspaces; other subspaces are all lines passing through the origin, and all planes passing through the origin. The lines and planes must pass through the origin since the zero vector 0 = (0, 0, 0) must be an element in each subspace.

Similar to Definition 3.1.1 we can define linearly independent vectors.

In R³, the vectors i = 〈1, 0, 0〉, j = 〈0, 1, 0〉, and k = 〈0, 0, 1〉 are linearly independent since the equation k₁i + k₂j + k₃k = 0 is the same as

k₁〈1, 0, 0〉 + k₂〈0, 1, 0〉 + k₃〈0, 0, 1〉 = 〈0, 0, 0〉 or 〈k₁, k₂, k₃〉 = 〈0, 0, 0〉.

By equality of vectors, (iii) of Definition 7.2.1, we conclude that k₁ = 0, k₂ = 0, and k₃ = 0. In Definition 7.6.3, linear dependence means that there are constants k₁, k₂, … , k_n not all zero such that k₁x₁ + k₂x₂ + … + k_nx_n = 0. For example, in R³ the vectors a = 〈1, 1, 1〉, b = 〈2, −1, 4〉, and c = 〈5, 2, 7〉 are linearly dependent since (3) is satisfied when k₁ = 3, k₂ = 1, and k₃ = −1:

3〈1, 1, 1〉 + 〈2, −1, 4〉 − 〈5, 2, 7〉 = 〈0, 0, 0〉 or 3a + b − c = 0.

We observe that two vectors are linearly independent if neither is a constant multiple of the other.

Basis

Any vector in R³ can be written as a linear combination of the linearly independent vectors i, j, and k. In Section 7.2, we said that these vectors form a basis for the system of three-dimensional vectors.

Standard Bases

Although we cannot prove it in this course, every vector space has a basis. The vector space P_n of all polynomials of degree less than or equal to n has the basis {1, x, x², … , xⁿ} since any vector (polynomial) p(x) of degree n or less can be written as the linear combination p(x) = c_nxⁿ + … + c₂x² + c₁x + c₀. A vector space may have many bases. We mentioned previously the set of vectors {i, j, k} is a basis for R³. But it can be proved that {u₁, u₂, u₃}, where

u₁ = 〈1, 0, 0〉, u₂ = 〈1, 1, 0〉, u₃ = 〈1, 1, 1〉

is a linearly independent set (see Problem 23 in Exercises 7.6) and, furthermore, every vector a = 〈a₁, a₂, a₃〉 can be expressed as a linear combination a = c₁u₁ + c₂u₂ + c₃u₃. Hence, the set of vectors {u₁, u₂, u₃} is another basis for R³. Indeed, any set of three linearly independent vectors is a basis for that space. However, as mentioned in Section 7.2, the set {i, j, k} is referred to as the standard basis for R³. The standard basis for the space P_n is {1, x, x², … , xⁿ}. For the vector space Rⁿ, the standard basis consists of the n vectors

e₁ = 〈1, 0, 0, … , 0〉, e₂ = 〈0, 1, 0, … , 0〉, … , e_n = 〈0, 0, 0, … , 1〉. (4)

If B is a basis for a vector space V, then for every vector v in V there exist scalars c_i, i = 1, 2, …, n such that

v = c₁x₂ + c₂x₂ + … + c_nx_n. (5)

The scalars c_i, i = 1, 2, … , n, in the linear combination (5) are called the coordinates of v relative to the basis B. In Rⁿ, the n-tuple notation 〈a₁, a₂, … , a_n〉 for a vector a means that real numbers a₁, a₂, … , a_n are the coordinates of a relative to the standard basis with e_i’s in the precise order given in (4).

Dimension

If a vector space V has a basis B consisting of n vectors, then it can be proved that every basis for that space must contain n vectors. This leads to the next definition.

EXAMPLE 5 Dimensions of Some Vector Spaces

If the basis of a vector space V contains a finite number of vectors, then we say that the vector space is finite dimensional; otherwise it is infinite dimensional. The function space Cⁿ(I) of n times continuously differentiable functions on an interval I is an example of an infinite-dimensional vector space.

Linear Differential Equations

Consider the homogeneous linear nth-order differential equation

(6)

on an interval I on which the coefficients are continuous and a_n(x) ≠ 0 for every x in the interval. A solution y₁ of (6) is necessarily a vector in the vector space Cⁿ(I). In addition, we know from the theory examined in Section 3.1 that if y₁ and y₂ are solutions of (6), then the sum y₁ + y₂ and any constant multiple ky₁ are also solutions. Since the solution set is closed under addition and scalar multiplication, it follows from Theorem 7.6.1 that the solution set of (6) is a subspace of Cⁿ(I). Hence the solution set of (6) deserves to be called the solution space of the differential equation. We also know that if {y₁, y₂, … , y_n} is a linearly independent set of solutions of (6), then its general solution of the differential equation is the linear combination

y = c₁y₁(x) + c₂y₂(x) + … + c_ny_n(x).

Recall that any solution of the equation can be found from this general solution by specialization of the constants c₁, c₂, … , c_n. Therefore, the linearly independent set of solutions {y₁, y₂, … , y_n} is a basis for the solution space. The dimension of this solution space is n.

EXAMPLE 6 Dimension of a Solution Space

The set of solutions of a nonhomogeneous linear differential equation is not a vector space. Several axioms of a vector space are not satisfied; most notably the set of solutions does not contain a zero vector. In other words, y = 0 is not a solution of a nonhomogeneous linear differential equation.

Span

If S denotes any set of vectors {x₁, x₂, … , x_n} in a vector space V, then the set of all linear combinations of the vectors x₁, x₂, … , x_n in S,

{k₁x₁ + k₂x₂ + … + k_nx_n},

where the k_i, i = 1, 2, … , n are scalars, is called the span of the vectors and written Span(S) or Span(x₁, x₂, … , x_n). It is left as an exercise to show that Span(S) is a subspace of the vector space V. See Problem 33 in Exercises 7.6. Span(S) is said to be a subspace spanned by the vectors x₁, x₂, … , x_n. If V = Span(S), then we say that S is a spanning set for the vector space V, or that S spans V. For example, each of the three sets

{i, j, k}, {i, i + j, i + j + k}, and {i, j, k, i + j, i + j + k}

are spanning sets for the vector space R³. But note that the first two sets are linearly independent, whereas the third set is dependent. With these new concepts we can rephrase Definitions 7.6.4 and 7.6.5 in the following manner:

7.6 Exercises Answers to selected odd-numbered problems begin on page ANS-17.

In Problems 1–10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set.

1. The set of vectors 〈a₁, a₂〉, where a₁ ≥ 0, a₂ ≥ 0
2. The set of vectors 〈a₁, a₂〉, where a₂ = 3a₁ + 1
3. The set of vectors 〈a₁, a₂〉, scalar multiplication defined by k〈a₁, a₂〉 = 〈ka₁, 0〉
4. The set of vectors 〈a₁, a₂〉, where a₁ + a₂ = 0
5. The set of vectors 〈a₁, a₂, 0〉
6. The set of vectors 〈a₁, a₂〉, addition and scalar multiplication defined by
   

   

7. The set of real numbers, addition defined by x + y = x − y
8. The set of complex numbers a + bi, where i² = −1, addition and scalar multiplication defined by
   

   

9. The set of arrays of real numbers , addition and scalar multiplication defined by
   

   

10. The set of all polynomials of degree 2

In Problems 11–16, determine whether the given set is a subspace of the vector space C(−, ).

11. All functions f such that f(1) = 0
12. All functions f such that f(0) = 1
13. All nonnegative functions f
14. All functions f such that f(−x) = f(x)
15. All differentiable functions f
16. All functions f of the form f(x) = c₁e^x + c₂xe^x

In Problems 17–20, determine whether the given set is a subspace of the indicated vector space.

17. Polynomials of the form p(x) = c₃x³ + c₁x; P₃
18. Polynomials p that are divisible by x − 2; P₂
19. All unit vectors; R³
20. Functions f such that f(x) dx = 0; C[a, b]
21. In 3-space, a line through the origin can be written as S = {(x, y, z) | x = at, y = bt, z = ct, a, b, c real numbers}. With addition and scalar multiplication the same as for vectors 〈x, y, z〉, show that S is a subspace of R³.
22. In 3-space, a plane through the origin can be written as S = {(x, y, z) | ax + by + cz = 0, a, b, c real numbers}. Show that S is a subspace of R³.
23. The vectors u₁ = 〈1, 0, 0〉, u₂ = 〈1, 1, 0〉, and u₃ = 〈1, 1, 1〉 form a basis for the vector space R³.
    1. Show that u₁, u₂, and u₃ are linearly independent.
    2. Express the vector a = 〈3, −4, 8〉 as a linear combination of u₁, u₂, and u₃.

24. The vectors p₁(x) = x + 1, p₂(x) = x − 1 form a basis for the vector space P₁.
    1. Show that p₁(x) and p₂(x) are linearly independent.
    2. Express the vector p(x) = 5x + 2 as a linear combination of p₁(x) and p₂(x).

In Problems 25–28, determine whether the given vectors are linearly independent or linearly dependent.

25. 〈4, −8〉, 〈−6, 12〉 in R²
26. 〈1, 1〉, 〈0, 1〉, 〈2, 5〉 in R²
27. 1, (x + 1), (x + 1)² in P₂
28. 1, (x + 1), (x + 1)², x² in P₂
29. Explain why f(x) = is a vector in C[0, 3] but not a vector in C[−3, 0].
30. A vector space V on which a dot or inner product has been defined is called an inner product space. An inner product for the vector space C[a, b] is given by
    

    

    
    In C[0, 2π] compute (x, sin x).

31. The norm of a vector in an inner product space is defined in terms of the inner product. For the inner product given in Problem 30, the norm of a vector is given by = . In C[0, 2π] compute and .
32. Find a basis for the solution space of
    

    

33. Let {x₁, x₂, … , x_n} be any set of vectors in a vector space V. Show that Span(x₁, x₂, … , x_n) is a subspace of V.

Discussion Problems

34. Discuss: Is R² a subspace of R³? Are R² and R³ subspaces of R⁴?
35. In Problem 9, you should have proved that the set M₂₂ of 2 × 2 arrays of real numbers
    

    

    
    or matrices, is a vector space with vector addition and scalar multiplication defined in that problem. Find a basis for M₂₂. What is the dimension of M₂₂?
36. Consider a finite orthogonal set of nonzero vectors {v₁, v₂, … , v_k} in Rⁿ. Discuss: Is this set linearly independent or linearly dependent?
37. If u, v, and w are vectors in a vector space V, then the axioms of an inner product (u, v) are
    1. (u, v) = (v, u)
    2. (ku, v) = k(u, v), k a scalar
    3. (u, u) = 0 if u = 0 and (u, u) > 0 if u ≠ 0
    4. (u, v + w) = (u, v) + (u, w).
    
    Show that (u, v) = u₁v₁ + 4u₂v₂, where u = 〈u₁, u₂〉 and v = 〈v₁, v₂〉, is an inner product on R².
38. 1. Find a pair of nonzero vectors u and v in R² that are not orthogonal with respect to the standard or Euclidean inner product u · v, but are orthogonal with respect to the inner product (u, v) in Problem 37.
    2. Find a pair of nonzero functions f and g in C[0, 2π] that are orthogonal with respect to the inner product (f, g) given in Problem 30.