5.7 Length and preserving it

Length is a natural notion in the real world, but it needs to be defined precisely in vector spaces. Using complex numbers complicates things because we need to use conjugation. Length is related to magnitude, which is a measure of how big something is. Understanding length and norms is key to the mathematics of quantum algorithms, as we shall see in chapter 10

5.7.1 Dot products

Let V be a finite dimensional vector space over R or C and let v = (v₁, v₂, …, v_n) and w = (w₁, w₂, …, w_n) be two vectors in V.

The dot product of v and w is the sum of the products of the corresponding entries in v and w.

v · w = v₁ w₁ + v₂ w₂ + ··· + v_n w_n

If we think of v and w as row vectors, and so as 1 by n matrices, then

v · w = v w^T.

The dot product of the basis vectors e₁ = (1, 0) and e₂ = (0, 1) is 0. When this happens for real vectors we say...