learningCurve  <- function(X, y, Xval, yval, lambda) {
  
  #   Generates the train and cross validation set errors needed
  #   to plot a learning curve
  #   
  #   error_train, error_val <- learningCurve(X, y, Xval, yval, lambda) 
  #   returns the train and cross validation set errors for a learning curve. 
  #   In particular, it returns two vectors of the same length (error_train and
  #   error_val). Then, error_train[i] contains the training error for
  #   i example (and similarly for error_val[i]).
  #
  #   In this function, you will compute the train and cross validation errors for
  #   dataset sizes from 1 up to m. In practice, when working with larger
  #   datasets, you might want to do this in larger intervals.
  #
  
  # Number of training examples
  m <- nrow(X) # Note here X and Xval have the same number of examples

  # You need to return fill this values
  error_train <- rep(0, m)
  error_val <- rep(0, m)
  
  # =========================== YOUR CODE HERE ==============================
  # Instructions: Fill in this function to return training errors in
  #               error_train and cross validation errors in error_val.
  #               The vector numex_vec contains the number of training
  #               examples to use for each calculation of training error and
  #               cross validation error, i.e, error_train[i] and
  #               error_val[i] should give you the errors
  #               obtained after training on i examples.
  #
  # Note: You should evaluate the training error on the first i training
  #       examples (i.e., X[1:i, ] and y[1:i]).
  #
  #       For the cross-validation error, you should instead evaluate on
  #       the ENTIRE cross validation set (Xval and yval).
  #
  # Note: If you are using your cost function (linearRCostFunctionReg)
  #       to compute the training and cross validation error, you should
  #       call the function with the lambda argument set to 0.
  #       Do note that you will still need to use lambda when running
  #       the training to obtain the theta parameters.
  #
  # Hint: You can loop over the examples with the following:
  #
  #       for (i in 1:m){
  #           # Compute train/cross validation errors using training examples
  #           # X[1:i, ] and y[1:i], storing the result in
  #           # error_train[i] and error_val[i]
  #           ....
  #
  #       }
  #
  
  
  
  # =========================================================================
  return(list(error_train = error_train, error_val = error_val))
}