than let be the space of linear functions of each linear function of can be written as xbwhich is the sum of the constant function and : xb moreoverthe two functions are orthogonal with respect to the inner product on hchi [ ][ ] where is column vector of zeros as subspaces of gboth and are again hilbert spacesand their inner products and norms follow directly from the inner product on for exampleeach function in has norm khkh kbkand the constant function in has norm | the modification of the regularized optimization problem ( where the constant term is not penalized can now be written as min (yi ( xi )) kgk ghc = ( xb kbk , ( which further simplifies to min where is the vector of observe thatin this caseas the optimal tends to the sample mean of the {yi }that iswe obtain the "defaultregression modelwithout explanatory variables againthis is convex optimization problemand the solution follows from ( xb yn ( with ( xb( (xx - ) (xn- ) ( this results in solving for from and determining from ( as precursor to the kernel methods in the following sectionslet us assume that and that has full (columnrank then any vector can be written as linear combination of the feature vectors {xi }that isas linear combinations of the columns of the matrix xin particularlet xawhere [ an ]rn in this case ( reduces to (xxn- xxn in ) (in - ) assuming invertibility of (xxn- xxn in )we have the solution (xxn- xxn in )- (in - )ygram matrix which depends on the training feature vectors {xi only through the matrix of inner productsxx[hxi ithis matrix is called the gram matrix of the {xi from ( )the solution for the constant term is bb - ( xx> ait follows that the learner is linear combination of inner products {hxi xiplus constantgt ( xb xx> = = ai hxi xi