ABSTRACT
We have seen how to view Cn as a space of vectors where you can study the algebra of addition and scalar multiplication. However, there is more you can do with vectors than just add them and multiply them by scalars. Vectors also have properties of a geometric nature. For example, vectors have length (some people say magnitude). This is the concept we study in this chapter. We are motivated by our experience in Rn, specifically R2. By the famous theorem about right triangles (remember which one?), the
length square of x = (x1, x2) is ‖x‖2 = x21 + x22 . It is natural to define the length of x as ‖x‖ =
√ x21 + x22 . However, when dealing with complex vectors,
we have a problem. Consider x = (1, i) in C2. Then, following the previous formula, ‖x‖2 = 12+i2 = 1−1 = 0. Unless you are doing relativity in physics, this is troublesome. It says a nonzero vector in C2 can have zero length! That does not seem right, so we need to remedy this problem. We can use the idea of the magnitude of a complex number to make things work out. Recall that if the complex number z = a + bi, then |z|2 = a2 + b2 = zz = zz. Let’s define ‖x‖2 = |1|2 + |i |2 = 1 + 1 = 2; then ‖x‖ = √2, which seems much more reasonable. Therefore, we define ‖x‖2 = |x1|2 + |x2|2 on C2. With this in mind, we define what we mean by the “length” or “norm” of
a vector and give a number of examples. Common sense tells us that lengths of nonzero vectors should be positive real numbers. The other properties we adopt also make good common sense. We can also use our knowledge of the “absolute value” concept as a guide.
