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1. warmUpExercise.m的實現如下：

  function A = warmUpExercise()
%WARMUPEXERCISE Example function in octave
%   A = WARMUPEXERCISE() is an example function that returns the 5x5 identity matrix


A = [];
% ============= YOUR CODE HERE ==============
% Instructions: Return the 5x5 identity matrix 
%               In octave, we return values by defining which variables
%               represent the return values (at the top of the file)
%               and then set them accordingly. 






A=eye(5);






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

end

2. plotData實現如下：

   function plotData(x, y)
%PLOTDATA Plots the data points x and y into a new figure 
%   PLOTDATA(x,y) plots the data points and gives the figure axes labels of
%   population and profit.


figure; % open a new figure window


% ====================== YOUR CODE HERE ======================
% Instructions: Plot the training data into a figure using the 
%               "figure" and "plot" commands. Set the axes labels using
%               the "xlabel" and "ylabel" commands. Assume the 
%               population and revenue data have been passed in
%               as the x and y arguments of this function.
%
% Hint: You can use the 'rx' option with plot to have the markers
%       appear as red crosses. Furthermore, you can make the
%       markers larger by using plot(..., 'rx', 'MarkerSize', 10);




plot(x, y, 'rx', 'MarkerSize', 10); % Plot the data
ylabel('Profit in $10,000s'); % Set the y?axis label 
xlabel('Population of City in 10,000s');




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

end

3. computeCost實現如下：

     function J = computeCost(X, y, theta)
%COMPUTECOST Compute cost for linear regression
%   J = COMPUTECOST(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y


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


% You need to return the following variables correctly 
J = 0;


% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.


J=sum((X*theta-y).^2)/(2*m)






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

end

4. gradientDescent實現如下：

    function [theta, J_history] = gradientDescent(X, y, theta, alpha, num_iters)
%GRADIENTDESCENT Performs gradient descent to learn theta
%   theta = GRADIENTDESCENT(X, y, theta, alpha, num_iters) updates theta by 
%   taking num_iters gradient steps with learning rate alpha


% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);


for iter = 1:num_iters


    % ====================== YOUR CODE HERE ======================
    % Instructions: Perform a single gradient step on the parameter vector
    %               theta. 
    %
    % Hint: While debugging, it can be useful to print out the values
    %       of the cost function (computeCost) and gradient here.
    %
temp0=theta(1,1)-alpha*sum(X*theta-y)/m;
temp1=theta(2,1)-alpha*sum((X*theta-y).*X(:,2))/m;


theta(1,1)=temp0;
theta(2,1)=temp1;


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


    % Save the cost J in every iteration    
    J_history(iter) = computeCost(X, y, theta);


end

end

5. computeCostMulti實現如下：

   function J = computeCostMulti(X, y, theta)
%COMPUTECOSTMULTI Compute cost for linear regression with multiple variables
%   J = COMPUTECOSTMULTI(X, y, theta) computes the cost of using theta as the
%   parameter for linear regression to fit the data points in X and y


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


% You need to return the following variables correctly 
J = 0;


% ====================== YOUR CODE HERE ======================
% Instructions: Compute the cost of a particular choice of theta
%               You should set J to the cost.






J=sum((X*theta-y).^2)/(2*m)


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

end

6. gradientDescentMulti實現如下：

   function [theta, J_history] = gradientDescentMulti(X, y, theta, alpha, num_iters)
%GRADIENTDESCENTMULTI Performs gradient descent to learn theta
%   theta = GRADIENTDESCENTMULTI(x, y, theta, alpha, num_iters) updates theta by
%   taking num_iters gradient steps with learning rate alpha


% Initialize some useful values
m = length(y); % number of training examples
J_history = zeros(num_iters, 1);
n=length(theta);
theta_t=theta;
for iter = 1:num_iters


    % ====================== YOUR CODE HERE ======================
    % Instructions: Perform a single gradient step on the parameter vector
    %               theta. 
    %
    % Hint: While debugging, it can be useful to print out the values
    %       of the cost function (computeCostMulti) and gradient here.
    %


  for j=1 : n
     theta_t(j,1)=theta(j,1)-alpha*sum((X*theta-y).*X(:,j))/m
     
  end




   theta=theta_t;




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


    % Save the cost J in every iteration    
    J_history(iter) = computeCostMulti(X, y, theta);


end

end

7. featureNormalize實現如下：

     function [X_norm, mu, sigma] = featureNormalize(X)
%FEATURENORMALIZE Normalizes the features in X 
%   FEATURENORMALIZE(X) returns a normalized version of X where
%   the mean value of each feature is 0 and the standard deviation
%   is 1. This is often a good preprocessing step to do when
%   working with learning algorithms.


% You need to set these values correctly
X_norm = X;
mu = zeros(1, size(X, 2));
sigma = zeros(1, size(X, 2));


% ====================== YOUR CODE HERE ======================
% Instructions: First, for each feature dimension, compute the mean
%               of the feature and subtract it from the dataset,
%               storing the mean value in mu. Next, compute the 
%               standard deviation of each feature and divide
%               each feature by it's standard deviation, storing
%               the standard deviation in sigma. 
%
%               Note that X is a matrix where each column is a 
%               feature and each row is an example. You need 
%               to perform the normalization separately for 
%               each feature. 
%
% Hint: You might find the 'mean' and 'std' functions useful.
%       




mu=mean(X);


sigma=std(X);


mu_t=repmat(mu,size(X,1),1);
sigma_t=repmat(sigma,size(X,1),1);


X_norm=(X-mu_t)./sigma_t










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

end

8. normalEqn實現如下：

     function [theta] = normalEqn(X, y)
%NORMALEQN Computes the closed-form solution to linear regression 
%   NORMALEQN(X,y) computes the closed-form solution to linear 
%   regression using the normal equations.


theta = zeros(size(X, 2), 1);


% ====================== YOUR CODE HERE ======================
% Instructions: Complete the code to compute the closed form solution
%               to linear regression and put the result in theta.
%


% ---------------------- Sample Solution ----------------------




theta=pinv(X'*X)*X'*y;


% -------------------------------------------------------------




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


end