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 measures how big something is. Understanding length and norms is key to the mathematics of quantum algorithms, as we shall see in Chapter 10, “From Circuits to Algorithms.” length

5.7.1 Dot products

Let V be a finite-dimensional vector space over R, 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: dot product product$dot ·`strong

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

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