Chapter 13

Algebra and Geometry of Rp

13.1 Rp

The notation, Rp refers to the collection of ordered lists of p numbers. The order matters.Thus (1,2,3) ̸= (3,1,2).

Definition 13.1.1 Define

Rp ≡{(x1, · · · ,xp) : x j ∈ R for j = 1, · · · , p

}.

(x1, · · · ,xp) = (y1, · · · ,yp) if and only if for all j = 1, · · · , p, x j = y j. When

(x1, · · · ,xp) ∈ Rp,

it is conventional to denote (x1, · · · ,xp) by the single bold face letter x. The numbers x j arecalled the coordinates. The set

{(0, · · · ,0, t,0, · · · ,0) : t ∈ R }

for t in the ith slot is called the ith coordinate axis coordinate axis, the xi axis for short.The point 0≡ (0, · · · ,0) is called the origin. Points in Rp are also called vectors.

Thus (1,2,4) ∈ R3 and (2,1,4) ∈ R3 but (1,2,4) ̸= (2,1,4) because, even though thesame numbers are involved, they don’t match up. In particular, the first entries are notequal.

Why would anyone be interested in such a thing? First consider the case when p = 1.Then from the definition, R1 =R. Recall that R is identified with the points of a line. Lookat the number line again. Observe that this amounts to identifying a point on this line witha real number. In other words a real number determines where you are on this line. Nowsuppose p = 2 and consider two lines which intersect each other at right angles as shownin the following picture.

2

6 • (2,6)

−8

3•(−8,3)

Notice how you can identify a point shown in the plane with the ordered pair (2,6) .You go to the right a distance of 2 and then up a distance of 6. Similarly, you can identifyanother point in the plane with the ordered pair (−8,3) . Go to the left a distance of 8 andthen up a distance of 3. The reason you go to the left is that there is a − sign on the eight.From this reasoning, every ordered pair determines a unique point in the plane. Conversely,taking a point in the plane, you could draw two lines through the point, one vertical and theother horizontal and determine unique points x1 on the horizontal line in the above pictureand x2 on the vertical line in the above picture, such that the point of interest is identifiedwith the ordered pair (x1,x2) . In short, points in the plane can be identified with ordered

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