5.1 R2 and C1
We looked earlier at the real plane as a set of standard Cartesian coordinate pairs (x, y) with x and y in R representing points we can plot. Now we give these pairs an algebraic structure so that if u and v are in R2 then so is u + v. Also, if r is in R, then rv is in R2 as well. We carry out the addition coordinate by coordinate. The multiplication by r, called scalar multiplication, is also done that way.
If u = (u1, u2) and v = (v1, v2),
| u + v | = (u1 + v1, u2+v2) |
| ru | = (ru1, ru2) |
Using the origin O = (0, 0) as the identity element, R2 is a commutative group under addition. R2 is a two-dimensional vector space over R. This is possible because R is a field.
Rather than considering them as pairs or points, we now call u and v vectors. I use bold to indicate a variable or a ‘‘point’...
