Let and be non-zero vectors in . What is meant by saying that and are linearly independent? What is the dimension of the subspace of spanned by and if they are (1) linearly independent, (2) linearly dependent?
Define the scalar product for . Define the corresponding norm of . State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality. Under what condition does equality hold in the Cauchy-Schwarz inequality?
Let be unit vectors in . Let
Show that for any fixed, linearly independent vectors and , the minimum of over is attained when for some , and that for this value of we have
(i) (for any choice of and ;
(ii) and in the case where .