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.
%
hx = X * theta;
J = 1 / (2 * m) * sum((hx - y) .^ 2) + lambda / (2 * m) * theta(2:end)' * theta(2:end);

grad(1) =  1 / m * X(:,1)' * (hx - y);
grad(2:end) = 1 / m * X(:, 2 : end)' *  (hx - y) + lambda / m * theta(2:end);


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

grad = grad(:);

end