5.1 R² and C¹

In section 4.2, we looked 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. 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, we say that r is a scalar. r u and r v are in R² as well. We carry out the addition coordinate by coordinate. The multiplication by r, called scalar multiplication, is also done that way. scalar R² C¹

If u = (u₁, u₂) and v = (v₁, v₂),

Using the origin O = (0, 0) as the identity element, R² is a commutative group under addition. With scalar multiplication by elements of the field R, R² is a two-dimensional vector space over R.

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” is a vector. When we plot a vector, we draw it as an arrow from the origin (0, 0) to the point represented...