def l2_rls_train(tr_data, tr_labels, regLambda, multiclass=False):
    """
    Add column of ones to tr_data, and calculate  the optimal
    weight vector by setting th egradient to zero and using
    the matrix notation form to calculate the regularised least square model

    data: type and description of "data"
    labels: type and description of "labels"

    Returns: type and description of the returned variable(s).
    """

    # This is just to be consistent with the lecture notes.
    X, y = tr_data, tr_labels

    nrOfRows = X.shape[0]

    # Expand X with a column of ones.
    X_tilde = np.column_stack((np.ones(nrOfRows), X))

    # if multiclass change tr_labels into a matrix
    if multiclass:
        newY = np.zeros((tr_labels.shape[0], tr_labels.shape[0]))
        for i, el in enumerate(tr_labels):
            newY[i][el - 1] = 1
        y = newY

    nrOfColumns = X_tilde.shape[1]
    # Compute the coefficient vector.
    if regLambda == 0:
        w = np.linalg.pinv(X_tilde) @ y
    else:
        inverse = np.linalg.inv(np.dot( np.transpose(X_tilde), X_tilde)  + np.identity(nrOfColumns) * regLambda)
        w = np.dot( inverse, np.transpose(X_tilde) ) @ y
    # Return model parameters.
    return w

def l2_rls_predict(w, data):
    """
    A summary of your function goes here.

    data: type and description of "data"

    Returns: type and description of the returned variable(s).
    """

    # This is just to be consistent with the lecture notes.
    X = data

    # add column of ones (biases) to given data
    y = np.column_stack((np.ones(X.shape[0]), data))
    # Your code goes here

    return np.dot(y, w)