## Inner Product, Orthogonal Bases, and Orthogonal Matrices

It follows from (I.10) that the cosine of the angle y between the vectors x in (I.2) and y in (I.5) is

Figure I.5. Orthogonalization. |

Definition I.13: The quantity x Ty is called the inner product of the vectors x andy.

IfxTy = 0,thencos(y) = 0;hence, у = n/2ory = 3n/4. This corresponds to angles of 90 and 270°, respectively; hence, x andy are perpendicular. Such vectors are said to be orthogonal.

Definition I.14: Conformable vectors x and y are orthogonal if their inner product x Ty is zero. Moreover, they are orthonormal if, in addition, their lengths are 1: ||x|| = ||y|| = 1.

In Figure I...

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