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.
By means of a sketch, give a geometric interpretation of the scalar product in the case , relating the value of to the angle between the directions of and .
What is a unit vector? Let be unit vectors in . Let
(i) for any fixed, linearly independent and , the minimum of over is attained when for some ;
(ii) in all cases;
(iii) and in the case where .