Vector Space

The totality of the 2-vectors generated by the various linear combinations of two independent vectors u and v constitutes the two-dimensional vector space. Sincc wc arc dealing only with vectors with real-valued elements, this vcctor spacc is none other than R2, the 2-space we have been referring to all along. The 2-space cannot be generated by a single 2-vector, because linear combinations of the latter can only give rise to the set of vectors lying on a single straight line. Nor does the generation of the 2-space require more than two linearly independent 2-vectors—at any rate, it would be impossible to find more than two.

The two linearly independent vectors u and v are said to span the 2-space. They are also said to constitute a basis for the 2-space. Note that we said a basis, not the basis, because any pair of 2-vectors can serve in that capacity as long as they are linearly independent. In particular, consider the two vectors [1 0] and [0 I], which are called unit vectors. The first one plots as an arrow lying along the horizontal axis, and the second, an arrow lying along the vertical axis. Because they are linearly independent, they can serve as a basis for the 2-spacc, and we do in fact ordinarily think of the 2-space as spanned by its two axes, which are nothing but the extended versions of the two unit vectors.

By analogy, the three-dimensional vector space is the totality of 3-vectors, and it must be spanned by exactly three linearly independent 3-vectors. As an illustration, consider the set of three unit vectors n o o

FIGURE 4,3

FIGURE 4,3

where each e,- is a vedor with J as its ith element and with zeros elsewhere/ These three vectors are obviously linearly independent; in fact, their arrows lie on the three axes of the 3-space in Fig, 43. Thus they span the 3-spacc, which implies that the entire 3-space (R\

in our framework) can be generated from these unit vectors. Tor example, the vector can be considered as the linear combination e\ + 2ei + 2ej. Geometrically, we can first add the vectors e\ and 2?2 in Fig. 43 by the parallelogram method, in order to get the vector represented by the point (J, 2, 0) in the x\xi plane, and then add the latter vector to 2e3—via the parallelogram constructed in the shaded vertical plane—to obtain the desired final result, at the point (1,2, 2).

The further extension to «-space should be obvious. The «-space can be defined as the totality of «-vectors. Though nongraphable, we can still think- of the //-space as being spanned by a total of« («-element) unit vectors that are all linearly independent. Each «-vector, being an ordered «-tuple, represents a point in the «-spaec, or an arrow extending from the point of origin (i.e., the «-clement null vector) to the said point. And any given set of n linearly independent «-vectors is, in fact, capable of generating the entire «-space. Since, in our discussion, each element of the «-vector is restricted to be a real number, this «-space is in fact R".

The w-space we have referred to is sometimes more specifically called the Euclidean n-space (named after Euclid). To explain this latter concept, we must first comment briefly on the concept of distance between two vector points. For any pair of vector points u and v in a given space, the distance from u to v is some real-valued function d — d(u. u)

with the following properties: (1) when u and v coincide, the distance is zero; (2) when the two points are distinct, the distance from u to v and the distance from v to it are represented

* The symbol e may be associated with the German word eins, for "one/'

by an identical positive real number; and (3) (he distance between u and i; is never longer than the distance from u to w (a point distinct from u and v) plus the distance from w to v, Expressed symbolically, d(u, v) =0 (for u = i1)

The last property is known as the triangular inequality, because the three points h, i.\ and w together will usually define a triangle.

When a vector space has a distance function defined that fulfills the previous three properties, it is eallcd a metric space, However, note that the distance d{u, u) has been discussed only in general terms. Depending on the specific form assigned to the d function, there may result a variety of metric spaces. The so-called Euclidean space is one specific type of metric space, with a distance function defined as follows. Let point u be the //-tuple

{Q[,ai an) and point v be the «-tuple (6i,/>2, ■ - then the Euclidean distance function is d(u, u) = \/(ai - b[)2 + (a2 ~ h2)2 + ■ ■ ■ + (an - bn)2

where the square root is taken to be positive. As can be easily verified, this specific distance function satisfies all three properties previously enumerated. Applied to the two-dimensional space in Tig. 42a. the distance between the two points (6. 4) and {3, 2) is found to be

This result is seen to be consistent with Pythagoras's theorem, which slates that the length of the hypotenuse of a right-angled triangle is equal to the (positive) square root of the sum of the squares of the lengths of the other two sides, Kor if we take (6. 4) and (3. 2) to be u and ?;, and plot a new point w at (6,2), we shall indeed have a right-angled triangle with the lengths of its horizontal and vertical sides equal to 3 and 24 respectively, and the length of the hypotenuse (the distance between u and v) equal to V32 + 22 = v^•

The Euclidean distance function can also be expressed in terms of the square root of a scalar product of two vectors. Since u and u denote the two «-tuples (ai an) and {b]%,. *, M, we can write a column vector u — l\ with elements a\ - b]< a2 — h2s,.., aK - ¿v What goes under the square-root sign in the Euclidean distance function is, of course, simply the sum of squares of these n elements, which, in view of Example 3 of this section, can be written as the scalar product lu - vY(u - tO- Hence we have d{u, u) — \J{u — i;)'(w — v)

EXERCISE 4.3

1. Given i/ = [5 1 3], ✓ = [3 T -!],*>' = [7 5 8], and x' = [x} x2 write out the column vectors, a, v, wt and x, and find

r 3i

_

_ -

r

2. Given w =

2

-

>Y =

yi

, and z-

_16_

(a) Which of the following are defined: wfx, xy(, xy\ y'y, zz\ yw\x ■ yl (ib) Find all the products that are defined.

3. Having sold n items of merchandise at quantities and prices PPn, how would you express the total revenue in (a) ^ notation and (b) vector notation?

4. Given two nonzero vectors w^ and w2, the angle 0 (0 < 6 < 180") they form is related to the scalar product w\w2 (= w^wi) as follows:

acute right obtuse angle if and only if v^ w2

Verify this by computing the scalar product for each of the following pair of vectors (see Figs, 4,2 and 4,3):

"3"

"1 "

2

=

4

"1 "

-3

4 _

, =

-2

"3"

-3"

2j

, w2 =

T

"0"

0

, W2 =

2

.0.

"l "

2

,tv2 =

2

, find the following graphically:

6. Since the 3-space is spanned by the three unit vectors defined in (4.7), any other 3-vector should be expressible as a linear combination of e\, ej, and eg. Show that the following 3-vectors can be so expressed:

7. In the three-dimensional Euclidean space, what is the distance between the following points?

(a) a 2, 8) and (0,-1,5) (b) <9, 0,4) and (2,0, -4)

8. The triangular inequality is written with the weak inequality sign <r rather than the strict inequality sign <. Under what circumstances would the part of the inequality apply?

9. Express the length of a radius vector vin the Euclidean n-space (i.e., the distance from the origin to point v) by using each of the following:

(o):scalars (b) a scalar product (c) an inner product

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