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.
%

real_theta = theta(2:end, :);

regularization = lambda * sum(real_theta .^ 2) / (2 * m);
prediction = sum(((X * theta) .- y) .^ 2) / (2 * m);
J = prediction + regularization;

mask = ones(size(theta));
mask(1) = 0;

gradient0 = sum(((X * theta) .- y) .* X) / m;
offset = lambda * (theta .* mask) / m;
gradient = gradient0 .+ offset';
grad = gradient;

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

grad = grad(:);

end
