linearRegCostFunction.m

function [J, grad] = linearRegCostFunction(X, y, theta, lambda)
%LINEARREGCOSTFUNCTION Compute cost and gradient for regularized linear 
%regression with multiple variables
%   [J, grad] = LINEARREGCOSTFUNCTION(X, y, theta, lambda) computes the 
%   cost of using theta as the parameter for linear regression to fit the 
%   data points in X and y. Returns the cost in J and the gradient in grad

% Initialize some useful values
m = length(y); % number of training examples

% You need to return the following variables correctly 
J = 0;
grad = zeros(size(theta));

% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost and gradient of regularized linear 
%               regression for a particular choice of theta.
%
%               You should set J to the cost and grad to the gradient.
%

 	%J=sum((X*theta-y)'*(X*theta-y))/(2*m)+lambda/(2*m) * sum(theta([2:end]).^2 );  % it also gives the correct output
 	J=sum((X*theta-y).^2)/(2*m)+lambda/(2*m) * sum(theta(2:end).^2 );



 	grad=(1/m)*X'*(X*theta-y)+(lambda/m)*[0;theta(2:end)]; % it also works.

 	%grad=(1/m)*sum(X*theta-y)*X' + (lambda/m)*[0;theta(2:end)]; % it returns the less value.
 		



   










% =========================================================================

grad = grad(:);

end