5.1 The scalar product in Rn普通地质学普通地质学.pdf

The scalar product in Rn Introduction We can add to the structure of a vector space by defining a scalar product or inner product (to every pair of vectors it associates a scalar rather than a third vector). In general, if V is a vector space with a scalar product, then two vectors in Vthen two vectors in V are said to be are said to be orthogonalorthogonal if their if their scalar product is zero. Scalar product or inner product are useful in many applications. For example, in the setting of a vector space with an inner product, we are able to solve general least squares problems. Introduction n In this section we define scalar product in R , 2 3 and we discuss results about scalar product in R or R including orthogonality and projections. n and show results about scalar product in R . Outline 1. Definition of scalar product in Rn 2. Results about scalar product in R2 or R3 3. Results about scalar product in Rn Definition of scalar product in Rn n The scalar product of two vectors x and y in R : x y     1 1     x2 n y2 n     xx   RR ,, yy   RR ,,         xn yn     T x y x y x y x y .