8. Show that if fxjgn j1 is a collection of mutually orthogonal, unit vectors in Rn, namely...

8. Show that if fxjgn j¼1 is a collection of mutually orthogonal, unit vectors in Rn, namely xj  xk ¼ 0 for j0k, and jxj j2 ¼ xj  xj ¼ 1 for all j, then for a vector y A Rn that can be expressed as a linear combination of these vectors y ¼

Xn j¼1 ajxj ; ð3:53Þ

the constants aj must satisfy aj ¼ xj  y. (Hint: Consider an inner product of each side with xj .)

Remark 3.49 The usual terminology is that the collection of vectors, fxjgn j¼1, are orthonormal. With the tools of linear algebra, it can be shown that all vectors y A Rn can be represented as in (3.53).